I was born in PR and my birth certificate has my fathers and mothers last name. I'm living in the states now and was wondering how do I go about to start this process.

Go to a local Family court and check the process. If you are an adult, it's a simple process.

Go to a local Family court and check the process. If you are an adult, its a simple process.

To clarify, we are talking about a name-change petition, right? I don't think OP can, or needs to, change the BC.

Yes, I don't see any reason to change the BC, not even sure why he wants a name change, but that's what he should ask for.

The BC does not change, but you get a court order changing your name, In Georgia, you go to the probate court, file out a form pay about 250 dollars and it is done in a short time.