Good, Evil, Law and Chaos are all as real as Magic
They are fundamental planar substances, like chaos diamonds or souls or rods of blasting or Githyanki.
You have to decide what is Evil in your world, if it's absolute, if it's intent-based, what the deal is with it. Ergo with Good. And Chaos. And Law.
And then any time anyone takes any of those actions, you have to increment their alignment a bit. As much as the action is towards that. If they're in the small segment of the population whose allegiance to one of the fundamental planar forces is high enough to shift them from 'Neutral', you tell them that. Otherwise they are Neutral.
How I handle it is to treat self-serving actions as Neutral - those which cost you something or are against your self-interest can be chaotic, lawful, good, or evil - harming someone if it costs you something is evil, helping someone if it costs you something is good, breaking the rules or changing the plan if it's not the best idea/costs you something is chaotic, following the rules or plan if you don't want to/it costs you something is lawful.
This is due to the description of the alignments in the PHB about self-serving interests being neutral and few people being anything but True Neutral. It works well for me, so that's the way I run it.
The Problem with Intent-Based Alignments
Very rarely do people think of themselves as evil, except due to guilt. They think their action is the correct one, even if it's nasty or self-serving. The person they murdered 'had it coming' or 'it was an accident' or 'it needed to be done' etc. So if it's intent-based, clearly evil people will not register as Evil, as they consider their actions as totally reasonable.
The Problem with Absolute Standards
Theft is always wrong is totally fine right up until the starving orphan steals an apple to give to his bedridden sister taking nothing for himself in the tiny gap in the roof they live in after running away from the orphanage run by evil matrons who murder and torture the children living there.
You can build ever more elaborate absolute structures, but there's always some situation that falls through the cracks. No absolute system of rules about what is 'evil' and what is 'good' ever covers every eventuality, and it's even worse for the more labyrinthine motivations of 'law' and 'chaos'.
How to solve this?
I use a combination of both, as I detailed above. You can also just plug this into modern day values - we see certain things as 'good' and certain things as 'evil' - using those values, the world can be strictly defined with little difficulty, and often, this is the default state of the world.
Evil in the Party
Most parties are formed of characters who consider themselves Good, often Saturday Morning Cartoon or Superhero good, Goody Two Shoes Good. As such, having an evil character around, unless that character hides their villainy, can be a source of friction.
Friction is not necessarily bad. People can disagree without necessarily murdering each other - as long as there's a reason to stick together, a mission whose importance overrides smaller disagreements, outright arguments and hatred can even be a good thing. They add interest, a LOT of interest, to the game and the characters.
Of course, it takes a level of roleplaying skill to portray this without taking it to extremes. Often, people will be murdering each other. Whether you want to allow that depends on you. You can initiate circumstances to forcibly separate the combatants somehow, such as via enemy attack, barmaid walking in, etc. Then if there is no reconciliation or way of party staying together, someone becomes an NPC, and that NPC becomes a recurring character working either with or against the party.
I find that players really appreciate having the choice to do actions that take them out of the plot. If so, in my games, something redirects them back to the plot (girlfriend they left with is killed by bbeg - king hires them to shadow party, they find out, he joins mission - they go alone to a monastery and pray and later have a divine revelation), or they simply become an NPC and the player rolls up a new character. It's not necessarily a bad thing - most stories have an evolving cast.
But having an Evil character in a Good party can work - Evil player characters can become bad guys for the party to defeat - and Evil is whatever you want it to be, whatever accentuates the story. That's all i'm really here to say.