And on characterizing evil in a deep, dynamic way: just like in real life, a villain is still a person. To really develop a compelling villain, it's generally recommended to focus on their perspective. Beliefs, reasonings, how they justify their choices to themselves.
Villains can be villains, but their appeal is that whether they're sympathetic or not, every memorable villain has identifiable motivations behind their villainy that make them such identifiable characters.
Lord Ozai isn't evil because evil is his motive, to him there's a logic to every evil he commits. Birthright, cultural glory, familial honor, political power, war strategy, his choices all revolve around reinforcing his own status. Most evil people, in and out of fiction, won't view their own motives as evil, nor even see the relevance. It's a means to an end, and often, to them, "evil" is just a slanderous opinion assigned arbitrarily to them by their opposition– competitors, haters, degenerates.
Even Junko Enoshima, who would claim she does do evil for evil's sake, was inspired to become a villain because she obsessed over the first thing that held her interest. She found something she cared about, and she latched her identity onto it.
So every multidimensional villain has a deeper motive. It may contextualize how or why they're evil. Plenty of villains think they're the hero, sure, and they'll believe they're the ones fighting for the greater good; but a lot of evils come down to acting out of self-service at the expense of others. And it's not always the case, but these motives can coexist too (ex: Vriska/Aranea).
The mod team and I have gotten on this topic before, and we went on to talk about Azula. Terrible person! But it hurts knowing she'd been suffering all that time too, and finally started cracking under the pressure when she inevitably drove away the only support that kept her marginally sane for a while. She's just like the rest of the Gaang when the hair's down and the lipstick's off, but it got in her head that she was better than everyone else, because she had to be, so she couldn't have equals, especially her own age. No leveling with her peers. Even friends are just assets. And that's a lonely way to grow up.
Khaos pointed out too how Ozai encouraged that way of thinking, using Zuko to set an example. It was always about surviving their father. And that was something she could never admit out loud, even after opening up about what her mother was "right" about, because at least she could still intimidate her peers with that confession. But the writing's on the wall. And like they said, Azula knows damn well that her grandpa died and her mom vanished on the same night all so Ozai could be firelord. She had no doubt her dad killed them, and she'd be next if she lost his favor.
Ozai would justify all of his villainy as discipline. Azula would rationalize all of hers as survival, though she wouldn't phrase it in such a vulnerable way. Zuko didn't stay a villain because he couldn't bring himself to defend his villainy anymore.
So when developing a villain, it'll help to think of evil as the conclusion after assessing their motives, how they act according to those motives, how they regard the consequences.
How they justify their actions? Do they recognize that they're doing anything wrong? Why or why not?