I personally think Azula is either a sociopath or she's borderline. She displays the following symptoms.
At least three are needed for a diagnose of antisocial personality disorder
1. Failure to conform to social norms with respect to lawful behaviors as indicated by repeatedly performing acts that are grounds for arrest;
2. Deceitfulness, as indicated by repeatedly lying, use of aliases, or conning others for personal profit or pleasure;
3. Impulsivity or failure to plan ahead;
4. Irritability and aggressiveness, as indicated by repeated physical fights or assaults;
5. Reckless disregard for safety of self or others;
6. Consistent irresponsibility, as indicated by repeated failure to sustain consistent work behavior or honor financial obligations;
7. Lack of remorse, as indicated by being indifferent to or rationalizing having hurt, mistreated, or stolen from another.
Azula consistently displays 1, 2, 4, 5, and 7.
Or she's borderline. (five or more are required of the following)
Any five of the following nine must be present.
1. Frantic efforts to avoid real or imagined abandonment.
2. A pattern of unstable and intense interpersonal relationships characterized by alternating between extremes of idealization and devaluation.
3. Identity disturbance: markedly and persistently unstable self-image or sense of self.
4. Impulsivity in at least two areas that are potentially self-damaging (e.g., promiscuous sex, eating disorders, binge eating, substance abuse, reckless driving). [Again, not including suicidal or self-mutilating behavior covered in Criterion 5]
5. Recurrent suicidal behavior, gestures, threats, or self-mutilating behavior such as cutting, interfering with the healing of scars, or picking at oneself.
6. Affective instability due to a marked reactivity of mood (e.g., intense episodic dysphoria, irritability, or anxiety usually lasting a few hours and only rarely more than a few days).
7. Chronic feelings of emptiness, worthlessness.
8. Inappropriate anger or difficulty controlling anger (e.g., frequent displays of temper, constant anger, recurrent physical fights).
9. Transient, stress-related paranoid ideation, delusions or severe dissociative symptoms
She displays 1, 2, 3, 6, 7, 8, 9