My Dungeon Master and I have been at arms for a week now since she allowed another player to kill me while I was away from the game, sick.

Now that I've built my new character, she tells me that I cannot make an evil character even though my background makes me evil.

She claims that my character isn't complete until she approves my background. I want to choose Dragon Casualty from the Optional backgrounds supplement to the Adventurers League season 4, Curse of Strahd (which is the campaign she is running).

Yes -- your DM can deny your background if she thinks it's going to make the game less fun. In the case of evil characters, there are strong reasons to think your background might make the game less fun.

D&D is built around having one adventuring party, meaning that all the player characters are in the same place doing the same thing at the same time. We have to do this, because there's only one DM and the DM can only tell one story at a time. So, when someone brings in a new character, it's expected that the group will make up reasons why the new character is friends with the rest of the group, and everyone trusts everyone else enough to adventure together.

(Adventuring with someone else is a lot of trust, because you sleep at night and expect your buddy to stay on watch for you! This means, in particular, you're trusting them not to kill you in your sleep.)

Even so, if someone creates a new character and the rest of the group says: "no, we don't like you, you can't adventure with us, go away" -- that would be really rude, basically equivalent to uninviting the player from the D&D group in real life! I've never seen a group do this.

When you declare that you want to play an evil character, does that mean that you're planning for there to be good reasons why the rest of the group can trust your character? Are you planning to be an upstanding and trustworthy member of the group?

Or are you planning to create a situation where they're socially obligated to trust you, even though they know it's a bad idea, and then abuse that trust by hurting them as much as you can?

There are other reasons to ban evil characters from Curse of Strahd. This adventure is all about the temptation to become evil, and fighting your character's gradual descent into madness. If you create a character who is already evil and enjoys it, the adventure doesn't really work so well.

Anyway: it sounds like it was pretty awful of your DM to allow some other character to kill your character while you were away. Your DM shouldn't have done that, unless maybe the other character had a really good reason. But letting you wreck the adventure in retaliation won't make the situation any better.

Your DM is right to ban your character.

You have two good options, going forward.

One option is to play a good or neutral character who is sincerely trying to work with the group and do the adventure. If you do this, you should have a conversation with your DM and with the other player; tell them you don't think killing your character is fun, and ask them to find a way to avoid this happening again.

The other option is, as KRyan says, to leave the game. If you don't think you can sincerely work with the group after they killed your last character, you're totally within your rights to tell the DM you're not coming back.

You should be strongly considering leaving this game—but over the other PC killing your character while you were away, far more than any concerns about an evil alignment being allowed or not. But as we often say, “no gaming is better than bad gaming.”

A group in which a fellow PC kills your character while you’re not even there sounds to me like it is always going to be bad gaming. And, for that matter, your head-butting over the alignment here suggests that this, too, is going to be a source of friction, even though it’s by a large margin the lesser issue.

To get it out of the way, ultimately, no one is forced to play the game, which is to say, no one is forced to play the game a particular way. The DM is not required to run a game they don’t wish to run, and a player is not required to play a game that they don’t wish to play. There is no other rule that takes precedence over this reality. So if your DM refuses to play a game with an evil PC, you can’t make them do so. If you refuse to play a game where you cannot be an evil NPC, they can’t make you, either.

What the books do or do not say is completely irrelevant at this point.

Because DMs tend to be scarcer than players, because DMing tends to be more work than playing, DMs usually have greater leverage in this regard, and for the record, the books largely are on their side. The Adventurers League does put some rules on the DM, but in non-AL play the books give the DM pretty much free reign. And, for that matter, the Adventurers League bans most evil alignments (you can only be Lawful Evil and then only when you belong to one of a couple of specific factions, which requires a particular background).

As the Adventurers League example shows, banning evil PCs is pretty common. Rather wrong-headed, in my opinion, because all alignments have serious risks of disrupting the game, and evil isn’t necessarily special in this regard, but nonetheless, it’s a pretty common thing to do. As a player, again, the only viable choice you get to make about this is to play or not play. That’s it. You cannot force the DM to allow something they do not want to allow.

Which brings us back around to the question of whether or not you should play: I’m pretty confident that the answer is no.

Banning evil-aligned PCs would be an irritation for me, as a player, and possibly a red flag that the DM either has fundamentally flawed beliefs about alignment in the first place or else simply doesn’t have a strong understanding of how it works out in practice, but nothing I’d quit a game over. But a PC killing another PC while the victim’s player is absent? No, sorry, thank you, but I have got better things to do with my time. That this happens suggests that the DM is, frankly, awful, and that the rest of the group is right there with them. I would want absolutely nothing to do with this group.

Real life happens. People cannot make every session. People cannot, even, necessarily give advanced warning that they will not make a session. As much as possible, you should make every effort to attend, and if you cannot, to give as much warning as possible—that’s only courteous, and regular failure to do so is a damn good reason to ask someone to leave the group. But it’s an out-of-game problem, that has to be addressed with an out-of-game solution. If they had a problem with you missing a session, that should have been a discussion they had with you—not something they “punished” you for in-character. “Punishment” has no place in a game. Anyone who thinks it does, is someone you don’t want to game with.

So I suggest you just give up on trying to make this DM happy, and try and find another game if you can. No need to fight about it, or attack them, or blame them—just a simple “you know, I think our ideas of what a game should be like are too different, and I don’t think this game is a good match for me or that I am a good match for this game. Best of luck, but I am bowing out.”

Yes, your GM can deny your Character

Take a look at your PHB:

Your DM might offer additional backgrounds beyond the ones included in chapter 4, and might be willing to work with you to craft a background that's a more precise fit for your character concept.^{PHB 5e p13}

Talk to your fellow players and your DM to decide whether your characters know one another, how they met, and what sorts of quests the group might undertake.^{PHB 5e p15}

This is what the Player Handbook tells you: you shall talk to your GM how your character ties into the campaign, and they tell you what is ok or what you should adjust. Note, that they talk about complete characters on page 15, and without a background (page 13!) you have no complete character. Without an allowed background, you are not ready to play.

The GM is entitled to do so because of the Dungeon Master's Guide:

As a referee, the DM acts as a mediator between the rules and the players. [...] Sometimes mediating the rules means setting limits. ^{DMG 5e p5}

The GM is the final arbitrator. They are there to set the rules and rulings of the game at the table, and the printed rules make the GM's word and decision higher than the printed rules. But to deny a character doesn't even need to crack out this whip but they can point to a different, equally binding rule that allows them to deny your character:

If they decide on campaign rules, those are binding. And these campaign rules include — so I read it from your description — that evil characters are not allowed for some reason. To set up such rules/restrictions is suggested by the DMG (emphasis mine). It clearly says that they are not only allowed but expected to set character generation rules as they see fit.

As you start to develop your campaign, you'll need to
fill in the players on the basics. [...]:

• 
  Any restrictions or new options for character creation,
  such as new or prohibited races. ^{DMG 5e p26}
  

Since their ruling is binding, you have to oblige.

What about AL?

Adventurers League allows Evil only as "Lawful Evil", so unless you are LE, your DM is even forced to deny your character in AL. But they only may accept your LE character if it conforms to all the AL rules - which disallows you to have the Background you want.

But this only concerns Adventurers League games under AL rules.

A way out?

Of course, you may leave or argue with them. Or you are constructive and try to get your DM to adjust the background to fit their campaign. Talk to them. Ask what you need to adjust. Indeed, the DMG tells the DM to talk back to you to adjust your character if needed.

Suggest alterations to a character's story so it better fits your world, or weave the first threads of your campaign into that story. ^{DMG 5e p26}