I was born outside the US as well. To get a copy of my birth certificate, I contacted the Bureau of Vital Statistics in the capital city of the country where I was born.

However, I should note that while I am a US citizen, I am also a citizen of the country where I was born. So the process may or may not be the same. I believe that if you are solely a citizen of the US but are born abroad, you apply through the State Department, but I could be mistaken about that.