Baby name and Birth Certificate changes

Kelsi • Under every pretty face is something complicated ✨💚

So, I need help. When my son was born me and my child’s father had issues so I named my son myself and the father didn’t sign my sons birth certificate. But now me and my child’s father have put the past behind us and I’m wanting to change my sons last name to the fathers last name and then add my child’s father to the birth certificate. How do I go about this?