Captain Leighton Perry is the captain of the Philadelphia Police Department. Like his subordinate, Lieutenant Carter Blake, he has a dislike to Norman, telling him in their first meeting that he doesn't need the FBI's help. He likes Blake's methods more than Norman's.
Perry happens to be more reasonable than Blake, as evidenced when he listens to Norman's lecture about the Origami Killer and allows him to continue on with it.
When Norman tries to tell him about Blake beating Ethan, Perry allows it to happen, rather than stopping it. He announces to the press that Ethan is the Origami Killer without his confession.
According to Norman, Perry cares about the press more than catching the Origami Killer.
If Blake kills Ethan while Norman is alive, he and Perry are suspended. For the latter's reason, it's either for bossing Norman around or declaring that Ethan is the Origami Killer without his confession.
If Norman dies while fighting the Origami Killer, Perry is forced to attend his funeral and allows Blake to not go there as both of them dislike Norman. When Blake asks him what to do with his ARI sunglasses, Perry tells him that he can do whatever he want with it.