As for Azula, she's the product of her parents just like Zuko. As you saw in the S2 flashback episode, the mother loved Zuko and knew that Azula was 'wrong' which of course made Azula even worse because her own mother was disgusted with her, so she just further idolized her father, who is also a power-mad sociopath.
But, see, that's the problem. She goes from evil to evil-er. She was evil, so her mommy didn't like her, and she became more evil because of it.
That's not satisfying at all for me. Sure, I suppose she could be idolizing her father, but I have seen no growth from her at all and simply idolizing her father in some abstract way (if that's what they've intended) isn't working. It seems she's just the embodiment of the 'Fire Nation people are evil, right?' sort of mindset that the writers have been very good about avoiding with Zuko, Iroh, etc.
During the S3 episode...
where a lot of character motivations are revealed at the beach she is the only character to basically have nothing to say--this struck me as either the writers shrugging their shoulders or putting a lantern on the fact that she is a bad plot device. Her two sidekicks were made better by this episode, revealing only further how terrible a character Azula is... It made a point at how badly she gets along with people... But, is her feeling of being an outcast supposed to justify her evil character traits?
I'm trying to remember the first introduction of her character and what I'm coming up with is Zuko feeding the Turtle-Ducks (Duck-Turtles?) and then showing his mother how Azula would feed them by dropping a rock on one. Was there anything at all more substantial, specifically that doesn't make her seem just plain crazy evil?
I don't know, maybe I'm over-criticizing the parts I don't like because so much of this show has been really well done.