Just wondering

Depends... would your family disown you if you did?

I don't think they would

I think it would be awkward.

That's just me though. You have to follow your heart.

It certainly can't be awkward in terms of being related by blood, since you aren't. Unless your cousin married a relative.
There are plenty of other ways, such as ages, or if either of you is married, or you are the Hatfields and McCoys, and so on.

It depends on your family dynamics, if they're close, if your own relationship with them is strong and healthy and you are well-regarded. It will also depend too if you do pursue the relationship with the niece as to whether it is firstly welcome, and if so, then how well the relationship goes. If you're confident that there's the potential for a loving relationship and your family understand that, it could be OK.

Wrong? I have no idea.

Illegal? No. Unless "going out with" is code for sex. And then only if she is under the age of consent.