-Tell your girlfriend the truth about you if she doesn't allready know, your past and that you're different now, whatever the hell that's supposed to mean.
-Tell her not to listen to this cousin of hers because he's just trying to get in your way for whatever reason, maybe he just doesn't like you.
-Just be cool about it, be cooool. Show that you are a really nice and decent guy so that this fucking cousin will eventually have to be ashamed of accusing you. If the rest of her family loves you because they see that you're a great guy who makes your girlfriend happy they will eventually turn on this foulmouthed asshole.
-Do not talk shit about the asshole, say nothing bad about him neither to your friends, your girlfriends family nor your own, just make sure your girlfriend understands that he's just wrong.
In time she will realize what is going on and so will everyone else, you just have to endure and stay strong. Teaching idiots a lesson(beating up, angrily confronting etc.) never works in the short run, and you don't have the situation here to wait for a long run to pay off. As I said just be cool and show by example.
It will work out fine if you do.