Anti-heroes always have their own beliefs and fight for what they stand for but they have questionable reasons. Anti-heroes have a lot of selfish actions and do not care about the consequences to their actions. Let’s just say that they have some kind of a moral compass.
They are independent, self-destructive and rebellious.You could use other references of many anti-heroes that’s popular.
Example: Jack Sparrow, Batman, Severus Snape, Harley Quinn, etc..
In conclusion, an anti hero is a complex character who fight for justice and equality in a villainous way. They have good intentions behind their ruthless actions, but they do have conflicts/struggles of themselves as well. They would go all above just to win an argument/fight, they usually value justice a lot. Remember that anti-hero are not the same with antagonist, but anti-heroes do good things but they do not use the same method that a hero would use.
I hope my explanation make sense.