TEEN REPORTER KAMILLE BOSTICK SAYS: Breaking up with someone because you don't like his cousin seems petty. You're dating the boyfriend, after all, not the cousin.
You're not clear as to why you and the cousin do not get along - maybe he's annoying, obnoxious, disrespectful, or you just don't like his hair. Whatever the reason, I can imagine you're constantly agitated that he's in your boyfriend's life and possibly in your vicinity, too.
The one thing you can't do is ask your boyfriend to choose between the two of you. His cousin/best friend have been that way a long time, and your demanding that something change probably won't be taken so well.
You seem to be ready to make a choice, but don't be too extreme. You have options other than breaking up.
First, simply try to avoid being around the cousin. I'm sure your boyfriend wouldn't mind keeping the two of you apart, especially since you both probably make him miserable always being at each other's throats. Let the cousin have his time with your boyfriend and you have your time. As often as you can, opt to not be around when the cousin comes around. That means declining invitations or just wrapping things up and leaving if the cousin comes around when you're with your boyfriend.
That said, there will be times when you and the cousin have to be around each other (say if your boyfriend plans a party and invites the two of you, or if you wanted to see your boyfriend and he happens to bring his cousin along).
In such cases, you're going to have to learn to put up with this cousin. Part of being in a relationship is making sacrifices. If this cousin isn't abusing you, insulting you or making you feel insecure or threatened, just ignore him. Learn to be civil. We don't always like everyone in our lives or jobs or classrooms, but we continue to function.
Ask yourself whether what your boyfriend gives you is worth your learning to deal with this cousin. If the boy is the greatest guy you've met, your best friend or just really fun to hang out with, then you shouldn't let this relationship go just because you aren't able to get along with the cousin. That gives the cousin far too much power in your relationship.
Take your power back and learn to look beyond this cousin.