The term you are looking for seems to be vindictiveness.
If you wanted to present this urge as a simple, non-directed feature of the character, then spite might be what you want: general-purpose destructiveness against anyone sufficiently vulnerable, or as Merriam-Webster has it, ‘petty ill will or hatred with the disposition to irritate, annoy, or thwart’. This would be an urge to harm anyone at all just because you can, and because that satisfies something in you. It is close to the idea of bullying or browbeating: doing harm for the sake of it, like Alex and his droogs celebrating ‘ultraviolence’ in A Clockwork Orange. The destructive act is the important thing: it doesn’t matter who the victim is.
However, if you want to discuss this urge as being in some sense motivated, or even (in the character's mind) justified, then vindictiveness is the term: ‘having or showing a desire to hurt someone who has hurt or caused problems for you’. This is an intent to harm a particular person for (in your mind) a particular reason.
It is close to the meaning of vengefulness, but that is more to do with settling a score or righting a specific wrong. Vindictiveness emphasises the desire to harm, within that.