The question is a bit unclear and I’m not quite ready to delete my other answer since I’m not sure it’s inappropriate, but I think this would get lost in it and may be closer to what Zach wants, based on comments.
Therefore, I am answering two questions here:
How should I tell a player that he’s not playing a character the way he should be played?
How should I deal with a player who describes his character as one thing, but plays him as something else?
I separate these two because they are different question, and need different answers.
1. How should I tell a player that he’s not playing a character the way he should be played?
Simple: you don’t. That’s none of your business. If a Paladin needs to Fall because of his actions, then that’s what happens (but see my other answer for thoughts on how best to handle that mechanically – I despise the official rules on the matter, and never recommend them), but it’s because he’s not being a Paladin, not because he’s not playing how you imagine the character. It’s his character, not yours.
In effect, when you ask how to do this without seeming to accuse him of bad roleplaying and/or trying to play his character, the answer is you can’t, because that is exactly what you would be doing. Maybe he is roleplaying badly; maybe that accusation isn’t unfounded, and maybe that’s a discussion you actually should have. But if he’s not roleplaying “badly,” there shouldn’t be any need to comment at all.
2. How should I deal with a player who describes his character as one thing, but plays him as something else?
This is a different matter. Particularly when the character’s backstory touches on the rest of the world, as your player’s does with his father as the head of an order of paladins, there is a greater cause for concern and a greater need to set things straight. Remember, while his character is his, the setting is yours. If your setting does not include any order of paladins that would condone or accept his behavior, then he can’t be a member of one – or he is about to get into trouble with them.
And if he was raised to be this way his entire life, and up until now upheld those standards, maybe they’re going to worry if he’s been cursed or possessed or something. But maybe he hasn’t been – maybe he felt stifled in the order, and now that he’s out in the real world he’s feeling a bit restless and rebellious. Maybe he’s looking for a bit of independence from a father in whose shadow he’d spent his entire life, whatever.
But the key thing is for the player to recognize that his actions do not jive with his backstory. I suggest that you give him these choices:
Change how you behave, to be more in line with how the character was described in the backstory.
Change your backstory, to make your character a bit more rebellious or a bit more light-hearted; maybe his father hoped that going out into the real world would make him a bit more serious.
Keep both the behavior and the backstory, and accept that this is very much an abrupt change in his behavior. Tell him that the order will not be amused, and will be greatly concerned about it, possibly even angry.
All of these things, however, have more to do with the order’s rules, and not the Paladin’s Code of Conduct. In both the second and third case, however, warn him that Falling is a very real concern for the order. They do not appreciate his behavior precisely because they believe it will cause him to Fall – and point out that putting others in danger needlessly is something that could cause him to Fall quickly.
But like I said in my other answer: a Paladin’s allowed to have a little, or even a lot, of harmless fun, so long as it is harmless, and he remains that unwavering bastion of Good and honor. Throwing a chair at someone or throwing an ally into a group of enemies may not be harmless, but there certainly are plenty of “un-Paladin-y” things he could do (e.g. harmless pranks) that will never cause a Fall.
There's no such thing as senseless violence, according to the one who commits it. Characters who kill or torture without at least an internal justification are crazy, not evil. You don't have a reason to kill people in the party or at random, so you don't. This doesn't make you nonevil.
Also remember that just because you're Evil doesn't mean you're a villain. Many Evil characters have no ambitions higher than their own survival and/or comfort; they don't aspire to great power, nor to purge the world of the target of their hate. They're just trying to get by, not so different from the rest of us.
The traditional list of Seven Deadly Sins was originally compiled not so much as a list of inherent sins, but a list of reasons that people sin. This makes it a great source of motives for Evil characters. I prefer to expand the list to nine, adding Fear as Wrath's twin in the fight-or-flight reflex, and Despair because it was actually in the original list; Sloth replaced it later.
Avarice: The key to happiness is having things. I will have it all.
Envy: I deserve it, not them. I will have it from them.
Gluttony: Pity those not at the top of the food chain. I will never be in that position.
Lust: I want to do it all, and I will let nothing get in my way.
Pride: I must be better then them: so much so that my superiority is never even questioned.
Sloth: I just don't want to do it. Let them do it for me. If they refuse, make them do it for me.
Wrath: They will never hurt me again. I will punish them for what they did, and leave them unable to do it to anyone else.
Fear: They must not be allowed to hurt me. (Note the lack of an again here: this is one of the big differentiators between Fear and Wrath, but it can make a huge difference in the character).
Despair: I just want the pain to end. Giving it to others helps.
Also keep in mind that these are core motivations. Any one of them will need to be elaborated upon. What is it? Who are they? How does the character plan to achieve this goal? Also worth noting is the lengths that your character goes to to hide her motives. Evil characters often prey upon one another's weaknesses, and while Wrath-type characters might not worry about seeming weak due to their motives, Sloth-type and Fear-type characters likely would. These folks are likely to construct a facade, often but not always based on Wrath, as a matter of posturing.
Your character sounds like a Wrath-type, with a focus on the undead. Because her main focus is on something that is not so amenable to the survival of humanity in general, she can get along decently well in society, and even be a very useful sort of person to have around. Some might even mistake her for heroic. But she has a twisted fight-or-flight reflex: any slight or injury, real or imagined, runs the risk of touching on that trauma, for reasons that make sense only to your character (if they even make sense to her). She might lash out disproportionately at small threats, or even against things she mistakenly believes to be threats, but are not.
Best Answer
From my understanding of the alignment in 5e, it is more representing your general behavior than the other way around.
So if you are more of a good human, then your alignment will be good. If you are more of an evil orc, then your alignment will be evil. If you are about 50/50, then you are neutral.
I would not foresee your alignment as swinging back and forth just because of a boost of anger here and there.