Well psychopathy is a term given for antisocial personality disorder. APD has the symptoms of:
Deceitfulness and compulsive lying. Azula demonstrates a readiness to lie whenever she believes it could give her even the slightest advantage.
Risk taking behaviors and a disregard for one's own safety. While Azula occasionally shows caution for her safety, it stems more from a very militaristic mindset, and she is much more willing to take a risk and potentially harm herself if it means getting what she wants.
Aggressive, manipulative, and antisocial behavior. Azula has a great talent for manipulating people and has a degree of pride in it, and is extremely aggressive towards those who don't acknowledge her authority. Antisocial in psychology means behavior harmful to socialization, as opposed to asocial, such as threatening and lying to friends, and attempting harm on one's family. When Azula has done repeatedly for the former and once in the comics for the latter.
Angry, bored, and discontent dispositions. Azula is naturally hostile to other people, and even uses her anger as a means of testing the water to how people act around her. When not on conquest or fight, she is constantly discontent with her surroundings and demands something to do. She wants action, even if it would be detrimental to her health or those around her.
Using charm or wit to deceive others for personal gain. See: recruiting Mai and Ty Lee.
Arrogance, a sense of self superiority and being extremely opinionated. These are core tenets of her character.
Repeatedly violating the rights of others through intimidation and dishonesty. See: interactions with her brother.
Lack of remorse of harming others. Cruelty towards animals (the turtle duck incident.)
Poor or abusive interpersonal relationships. See: how she treated her friends and thus abandoned her. How she treated her brother.
Seems clear to me, man.