I have an experienced player wanting to create a tempest cleric who doesn't worship a god. His back story will be something along the lines of 'walking through a storm one day he is hit by lightning, which provides him with power'.
I'm happy to entertain the thought, but what aspects might go awry?
The PHB (see Appendix B) notes that it is perfectly acceptable for a DM to decide that there are no gods that are worshipped in his or her campaign:
Your DM determines which god, if any, are worshipped in his or her campaign.
Emphasis mine. In these instances where the players inhabit a faithless world, the clerics, druids, and paladins don't just disappear because there are no imaginary sky people to give them super powers. They draw their power from a devotion to whatever ideal they value above all others.
The DMG goes on to codify this a little more explicitly, and provides guidance on alternative religious systems to the standard polytheistic pantheon that the vanilla game assumes as the default. It even specifically discusses "forces and philosophies" as an alternative to deities entirely on page 13. It has a decent amount of text, so instead of quoting it in its entirety, I'll simply leave it to you to look up at your leisure. You might also consider animism, which the DMG discusses on page 12. Perhaps that storm was caused by an Air Spirit, and the act of walking through the storm without praying to it for safe passage angered the spirit, and now it calls the character to its service. There are many possibilities that are well-supported by the rules and do not require any substantial effort on your part to implement.
It is simple. You may not have chosen a god but the god choose you. Your player may think he got his power by being struck by lightning, but maybe it is how the god chooses his followers.
Most spells can be rewritten to instead of calling to a god, he calls to his inner self(spoilers: which is calling to his god). This way none of the spells need to be redesigned. The player just thinks hes doing it all himself.
Mechanically speaking, there's no real distinction between arcane and divine spellcasting. If you wanted to, you could rewrite the text of varying class features so that they don't require the intervention of a god. It'd take a bit of contorting to explain how the cleric got the power to turn undead and access divine intervention, but it won't break the game.
The potential issue in this is not in any rule mechanics, but in the cleric's play-balance.
By the question "what aspects might go awry?" I am going to assume you mean "what issues will I have to manage either in setting up a character or in what might happen during play from the House Rule that clerics can gain their divine powers from a non-deity source?". I'm going to refer to the source of power for the character you have described as "the weather".
I believe that very little, if anything, is needed in terms of the mechanics of the rules as you would be treating "the weather" as if it were a deity. An obvious example of this working is the Druid, where the divine power source can be "the force of nature".
Power of Nature (PHB p.64):
Druids revere nature above all, gaining their spells and other magical powers either from the force of nature itself or from a nature deity.
Mechanically this concept would work for all the cleric's powers, even Divine Intervention (a discussion point in other answers) as it is down to the DM what actually happens:
Divine Intervention (PHB p.59)
The DM chooses the nature of the intervention; the effect of any cleric spell or cleric domain spell would be appropriate.
I think the real issue to consider is a cleric's core class concept of being a divine agent, granted power to further the cause of their deity in the mortal world. The cost of the divine power bestowed upon them is to be devoted to the deity's cause and principles, manifesting in the role playing of the character. This is actually a really important part of the cleric's play balance as it gives the DM a clear and rules based mandate to sanction a cleric for "bad" behaviour and ultimately strip them of their divine abilities if the character continues to defy their deity. It means a cleric character can't act how it wants whenever it wants, the powers come at a cost.
A house rule to allow a source of divine power with no such agenda effectively removes this core restriction and could significantly effect game balance, as the character is granted it's divine powers "for free" compared to other clerics. It occurs to me that this character would effectively be a divine warlock using the cleric class for the rules.
Whether this is an issue is down to how you run your game, how much your player will exploit the house rule and whether your troupe is comfortable with the play balance of the character. You may also need to consider what happens when the next player comes up with a character using the house rule that effects play balance even more?
An obvious option to manage this would be to give "the weather" an agenda and the will to impose this agenda on its clerics, but that makes it a deity in every way that matters and makes this question void.
I think it's an interesting character concept. From a DM's perspective I would make it clear that while "the weather" is not a deity as such, the character will have requirements on it's behaviour in the same way that "the force of nature" does on a Druid, just as inconvenient as if they worshiped a deity.
The question, "what aspects may go awry", is worth answering twice within two different frames of mind:
Nothing will go awry. In a pure, vanilla campaign on the Material Plane where there is this villain with some plot and your players set out to foil it, nothing written about how the class functions is going to come into play. He's going to cast the same spells, the opponent will make the same saves, levels will work as written and uninterrupted, and nothing will inherently change.
Everything could go awry. If you and your players are the type to really dig their claws into the story and these characters get into epic-level campaigns where they're jumping from plane to plane and usurping thrones from gods, things have the potential to get hairy. I've never run campaigns of this sort, but I can't imagine prayers from Pelor are well heard from the Nine Hells. As a DM I know I would want to play that up in some capacity — by the book or otherwise. Point being, casting divine spells without attaching them to the divine is a completely different kind of magic that would be affected (and not affected) by different circumstances that could very definitely arise.
For your specific scenario, I seldom give my players a flat "no" — especially when it comes to character creation — and I would let him create this character as he described and truly let his magical abilities come from within. I'd treat the nature of his spellcasting more like a sorcerer than I would a cleric, and would just have to make judgement calls on the fly if anything more than that came up.
Classically, Clerics get their power from deities, but you are free to reflavor things however you want. It might be that this Cleric gets their powers from a deity of nature, who chose to give them their power, even though the Cleric doesn't follow them. Or maybe the storm unleashes something inside of them. Alternatively, Paladins follow Ideals and Oaths, even though they are granted their powers from deities. Maybe this PC is technically a Paladin who just happens to have all of the class features of a cleric, and none of the features from a Paladin.
If you do, be sure to figure out how the level 10+ Cleric Feature Divine Intervention will work. Is is a chance to call upon the forces of nature to intervene in some way? Just make sure you match all the benefits of a deified Divine Intervention with your natural one.
That being said, I would see if your player wants to play a Storm sorcerer. IMO, that's more in-line with the backstory you have presented.
