You mean Harley Quinn with the Joker?

More like a female assassin becoming obsessed with a character who was guarding one of her marks and almost prevented her from completing her contract. She then goes on to stalk the character relentlessly, "attempting" to assassinate him, but purposefully messing up so that he has to interact with her. When she finally joins his group full-time, she basically says that she's, "the only one who gets to kill him".

The subject of her obsession eventually finds her obsession endearing, but otherwise is a normal, well-adjusted murderhobo. I mean, they still find her to be strange, and it takes them time to come to the realization that assassin girl isn't actually trying to kill them.

Ever see Death Note?

Yeah, but Light was hardly well-adjusted.

Yes its fine to show it. Just make sure that it is portrayed the way you want it to be. Get beta readers and second opinions to see how others take your writing. Characters should have flaws and should get in trouble if they didn't whats the point.

Yes, makes sense to me