I'm a very new DM, and I've played three games in total with a new group each spanning about thirty minutes (Role playing games aren't appreciated where I live) . One of my characters is a Chaotic neutral but he plays as a Lawful Good character. Is it okay for me to tell him how to roleplay or should I let him find his character on his own?
Don't tell him how to roleplay
Doing so would be a vast overstep in a social circle, as it would be telling someone how to play the game (taking control over the one thing they should have total control over).
As a GM, you can tell him his character's alignment is changing
While there is a lot of discussion and disagreement about alignment, almost everyone agrees on one thing: alignment doesn't determine actions, alignment emerges from actions. Alignment shift is a real thing - due to misunderstanding the alignment system (and accidentally roleplaying a different alignment), due to not knowing how you want to roleplay the character when picking those two letters, due to motivations shifting, due to traumas or interactions, etc.
I discourage just telling him "hey, you've been playing as LG, so you're LG now." More finesse and warning is needed - communicate. Talk with the guy, say "hey, I've noticed [character name] has been pretty altruistic and orderly. Do you think Chaotic Neutral is the right alignment? He's been a lot closer to Lawful Good recently, he might start shifting towards that if he keeps acting like it." Note that many mechanics don't care what alignment you are, so a large shift usually doesn't really impact the core "stuff" of the character (only a few effects, and which deities you can call on).
TL;DR: Don't tell your player how to roleplay their character, discuss whether their actions are in line with their alignment (and talk about adjusting alignment to fit the actions - actions lead to alignment).
Descriptive vs Prescriptive
Seeing alignment as descriptive means the alignment describes how a character in general reacts. Some characters tend to exploit or protect weak characters or uphold the law even though it is really inconvenient for them or break it just to screw with authority. Alignment is a general crude reading of a character's personality which is actually much more complex than 9 categories. Characters are free to act however they want, the alignment is decided based on past actions.
Seeing alignment as prescriptive means the character is preprogrammed to act a certain way in certain situation. It is just in their nature and they cannot help it.
Not everyone sees alignment the same. Trying to figure out which alignment various real or fictional characters have will tend to create debates that don't end up in everyone agreeing. If you can't even agree on which alignment Spiderman, Conan the Barbarian and Gandalf have there is little chance you can agree on the PC's alignment.
There is no right and wrong how to handle this. Your group just needs to set up ground rules (ideally in a session zero) about how alignment works in this world and what to do when players break their character's alignment.
You can force them to play differently, you can change their alignment or you can ignore alignment for most of the game, don't let them find the Sword of Goodness that only Lawful Good characters can attune to and make a ruling when it does come up.
It is, however, important that you make clear how alignment works in your game or to have the group discuss how it should work so everyone agrees.
I am reminded of a story told by my first DM:
I had a game where a thief (we'll call him Perfidy Quisling) was a great team player. He befriended other characters in-game, voluntarily pushed into the front lines, and risked himself to save the weak. He donated to beggars, made offerings to churches, and was a great all-round guy. I repeatedly took him aside and told him, "Hey, man, you are a great player but you aren't respecting your alignment [neutral evil]. If you keep playing like this, let's change you to lawful neutral or lawful good."
The player always told me, "Please don't change my character. I just play a little differently."
After more than three months of campaigning together like this several times per week, after the party had amassed considerable wealth, he wrote a short note to the DM while his character was on watch: "I quietly slit the throats of my party members, load the wagon with their valuables, and ride to the nearest town with the terrible news of our party's demise."
And so ended the game for his party.
As DM, you can tell players whatever you like, but do so privately, and make a judgment call based on the player's experience and history.
You ask about dnd-5e so why not look at what the rules actually say?
A typical creature in the worlds of Dungeons & Dragons has an alignment, which broadly describes its moral and personal attitudes. Alignment is a combination of two factors: one identifies morality (good, evil, or neutral), and the other describes attitudes toward society and order (lawful, chaotic, or neutral).
... Individuals might vary significantly from that typical behavior, and few people are perfectly and consistently faithful to the precepts of their alignment.
Alignment in the Multiverse
For many thinking creatures, alignment is a moral choice. Humans, dwarves, elves, and other humanoid races can choose whether to follow the paths of good or evil, law or chaos.
Alignment is an essential part of the nature of celestials and fiends. A devil does not choose to be lawful evil, and it doesn't tend toward lawful evil, but rather it is lawful evil in its essence. If it somehow ceased to be lawful evil, it would cease to be a devil.
The overarching concept is one of choice (unless you are a devil etc.); to my mind, the alignment written on a character sheet is aspirational - this is the type of person the character wants to be; not who they are.
As to specifics:
Lawful good (LG) creatures can be counted on to do the right thing as expected by society.
Chaotic neutral (CN) creatures follow their whims, holding their personal freedom above all else.
It occurs to me that the current "whim" of the character may be "to do the right thing as expected by society." From the outside, the Lawful Good and Chaotic Neutral individual are indistinguishable. From the inside the Lawful Good individual is acting in accordance with their moral compass and will consistently act that way, the Chaotic Neutral individual is just going with the flow: perhaps its easier to just move along when the guard asks rather than picking a fight. Of course, maybe the Chaotic Neutral will feel differently about it tomorrow but the Lawful Good won't.
And then, of course there is the rule (PHB p.6):
How to Play
- The DM describes the environment
2. The players describe what they want to do
3. The DM narrates the results of the adventurers’ actions
The DM narrates results - what effect the character's actions have in the world: the DM controls the world - including what the world thinks the character's morality is. The player describe what they want to do: the player controls the character - including what the character thinks their morality is. These do not have to match!
You'd really need to set up a lot of things beforehand, though, so you probably can't do this in your ongoing game.
Now, you don't have to do this: there are a lot of ways to play where this is a bad idea. But you can play in a manner where telling your players that they are breaking alignment is fine, even to the point of prohibiting some actions on the basis of their violating alignment. Here's how to do that:
Define alignments rigorously. Don't feel constrained by the general hand-wavey examples in the books-- pick ethical stuff sufficient to set up two axes with two opposing ends and then stick with it. For example, I sometimes run a Pathfinder game where Deontology ≡ Good and Utilitarianism ≡ Evil, while philosophical parsimony ≡ Law whereas a lack of parsimony/'existentialism' ≡ Chaos. This works pretty well and results in a world that fits intuitions pretty well, but I could just as easily have defined each of those things to be their opposite (and sometimes do for a one-off campaign) or use risk-taking preference as an axis (e.g. Law ≡ choose the certain option in a situation with identical average outcomes, Chaos≡ choose the highest potential output option in a situation with identical average outcomes) or any number of other things.
The key is to pick something that can be easily used as a spectrum with two discrete ends, and whose ends are both sufficiently rational near the meridian that actors acting in accordance with them don't automatically seem insane (so the principle of Benevolence is a bad map for Good, for example, because it makes Evil and even Neutral people largely incomprehensible/insane). It's also good to pick systems you have a decent command of, and feel comfortable passing judgment on test cases for. You can use Trolley problems(an example is linked) to help prep for such judgement passing.
Consider how Archons/Angels, Demons, Devils, and Eladrin work, what Celestia, the Abyss, Hell, and Arborea are like and why, and how various outsiders would explain why their system is the best system. For example, in the primary system outlined above, Asmodeus is a Utility Monster, so the only reasonable course of action is, of course, to give him all of the things. And indeed, he enjoys your suffering far more than you are upset about it, as do his many legions of devils. Greatest happiness for the greatest number of people and all that. (related comic)
Decide how you are going to handle alignment breaches and changes. What happens when an NPC changes alignment? What happens when a player changes alignment? How do players change alignment? What do you do when a player tries to do something counter to their alignment? etc. For example, you might give characters advantage on rolls for actions that exemplify acting according to their principles in defiance of external disagreement, and disadvantage on rolls for actions that are against the character's conscience. Alternatively, you might just require a saving throw when a player attempts to have their character break alignment, or any number of other things.
Explain to your players how each alignment is going to work, and how alignment in your game is going to work overall. Get player buy-in before character creation, and talk about how you are going to be adjudicating alignment, and what happens when players have their characters break alignment, and what you'll do when people are breaking alignment accidentally. Stick to these agreements while playing the game, as potentially modified by further discussion.
This system works fairly well, though you'll need to experiment to find the right sorts of things to use as axes for your group's enjoyment, and you'll probably pick some bad answers to 'what happens when a PC/NPC breaks alignment' til you get more experience. I strongly recommend erring on the side of leniency here-- the advantage/disadvantage thing works okay and you can just require players to fictionally justify alignment changes rather than punishing/rewarding them for them. But, in any case, prohibiting player action on the basis of alignment can work, as long as a solid alignment framework has been built first.
As the GM, you are in charge of the game and largely in charge of the players' enjoyment of it. You can and ought to open discussion on any matters of player characters' personality that you perceive might harm the enjoyment of the player or the group as a whole, and serve as the ultimate arbitrator on what steps to take when there is a social problem. Outright telling another player how (not) to play their character is a harsh, extreme measure, and I wouldn't do it without a strong consensus among the rest of the players - a milder solution to the problem you see here would be asking the player to mark their alignment as Lawful Good if they keep playing their character as such.
However, personally I would just let the player play their character as Lawful Good as they like without raising an issue about it. Unlike earlier editions of DnD, alignment in DnD 5e does practically nothing. For most intents you can simply pretend the alignment value doesn't exist, and rule based on the player's behavior if you encounter one of the very few situations where it really does matter. While you may be puzzled about the reasoning behind the player's conflicting choice of alignment and behavior, it will not cause any actual problem for your game in itself and thus you don't have a reason to interfere.
But that doesn't come close to scratching the surface of the iceberg this topic is, as evidenced by the plethora of passionate answers.
Alignment is Controversial
Anyone who discusses D&D with frequency knows that not everyone agrees on where the lines are drawn. How people of any given alignment act is exceedingly unclear, to the point of being almost meaningless, IF you try to go by what others tell you online.
To this end, you should talk to your players about what alignment means in your setting, and how various alignments impact characters and the party.
Players don't always pick their character's real alignment.
I've had players describe a concept to me, then upon greater discussion, it turned out that they really wanted to play a different alignment, but didn't want to write that other alignment on their sheet.
Sometimes this is down as a sort of "poser" bad-guy-cred, ala Green Ranger, where the PC will never actually be evil/chaotic in the actual game, but the player wanted to the feel of street-cred. While I can appreciate the goal, this is the wrong tool for that job.
Sometimes this is because players just really don't understand alignment. This is tough to identify, but hopefully, self-adjusts over time.
Don't use alignments
Honestly, this is the simplest approach. Odds are, alignment will spend more time getting in your way than helping you, so toss it.
Use goals and compatibility
Require your PCs to justify their cooperation, in a way that not only makes sense but is more permanent than your campaign. Have them be related(compatibility), wanting to stay part of the same guild(goals), or all loyal to the same Emperor(compatibility). As long as characters have predictable willness to be in the party, players are VERY unlikely to cause fights over alignment-esque my-guy issues.
Refuse character archetypes that don't fit your story/group.
Not only are you within your rights to cull incompatible players, it is in your best interests. If you're running a group of brazen pirates, wanting to sail the Grand Line in search of One Piece, tell your players: No allegiance-minded Paladin-types. If you're running siege-defense story where humanity is struggle to survive within the great walls of a massive city assaulted from without by giants, obviously, no low-commitment self-interested wanderers.
I love using mechanics to inform the players. You can set up scenarios that are only completed by the given alignment, knowing (and preparing for) their failure given the person will act differently.
"You can only pick up this hammer if you're pure of heart. Wait, you can't pick it up, apparently the hammer thinks you're an *******."
Even going so far as to have a spirit come and admonish the hero, pointing out how their actions and words have belied their actual nature.
This can spin off a whole new quest of discovery for the player and help them grasp who their character is and how they function. Even if the character is someone like the player, it can be interesting to explore our faults from a more-or-less third-party perspective. It also allows for adventure, as those trials to get back on the good side of the magical hammer can be harrowing (to say the least).
Other players can also benefit, as having to help the tormented player can help them explore how they feel about others and how they interact as a group when with the player or each other in some odd situations. I used a redemption story to drive a whole campaign, and was able to spin that off into an adventure for each player's character, making a fine campaign of it where everyone could hold the "center spotlight" for a few sessions and fully explore motivations. Even after that, the characters gained, well, character and future sessions with them were like visiting old friends. (Even when they all retired and ran weapon shops ^_^)
This is one of the things that Inspiration is for
Not all DMs award Inspiration, but your question describes one of the situations in which it might prove useful. You could chose to "use the carrot, not the stick," by awarding other players Inspiration for doing things that represent their alignments particularly well.
This fits well with how Inspiration is explained in the source books. Consider that in the PHB (p. 124), on "Gaining Inspiration", we read (emphasis mine):
Typically, DMs award it when you play on your personality traits... and otherwise portray your character in a compelling way.
Although that doesn't mention alignment explicitly, it does say "personality traits", and on p. 122 of the PHB, under "Alignment", we read (emphasis mine):
A typical creature in the worlds of Dungeons & Dragons has an alignment, which broadly describes its moral and personal attitudes.
So, Alignment is part of one's personal attitudes, and when a player especially exudes their personality traits (such as personal attitudes), then that is one of the cases where a DM would "typically" award Inspiration.
When doing so, you can say what it is for. "Hey, that was so iconic of your Lawful Evil alignment, I am awarding you Inspiration!" And that can get the attention of all the players, including those not playing to their alignments.
"Use the carrot, not the stick"