Azula was born a fast learner and a protege in firebending. Zuko was born average, but compared to his younger sister it made him look worse by comparison. Therein lies the tragedy of Azula. Ozai, as both a man and a father, craves heartless perfection in all things under his dominion. Azula's talent showed the potentiality for such a perfection, it made her Ozai's favorite. That is a position no one should ever be in.
Ozai prioritized Azula, bit by bit and year by year he fed his world into her young mind. Ursa found her kinship in the runt of the little, the fuck up, Zuko. Ozai took Azula under his wing, so Ursa did in kind with Zuko, like many bad marriages she began a division of what is "hers" and what is "his". Azula was his. And so Azula lost whatever opposing viewpoint she may have had in her tender years. She learned Ozai's cold ways, his competitive, Darwinist view of the world. She learned all of life as a challenge, all people are only allies if you rule over them, and those you can't are enemies. She learned there is no such thing as "good enough" that victory is the only acceptable condition. She learned, in a sibling's way, to loathe her brother and the mother that preferred him. She learned to take her mother's absence and hostility as a sign she was wrong.
The worst of the tragedy, in the end, is that Ozai raised a good right hand, not a successor. Azula without exception exceled above and beyond when presented with a goal or a task to master. She killed the avatar and took the unconquerable city. But when her father left her and said "rule" she was lost. Aimless. She turned inward and found herself alone with only herself as company. So focused on the competition of her father's war-world she did not realize that this wasn't good company. She snapped like a twig. And she will never be made whole again.
The broken once-princess of a would-be empire. Azula will die alone, without ever quite understanding why.