How do you add the father's name to a birth certificate in Colorado?

You go to court and request a modification of the birth certificate. He will be required to sign an acknowledgment of paternity and possibly take a DNA test.

It depends on why the name was left off in the first place. Also how old the child is. If the child is newborn, and the father is willing to sign, then you should be able to just go to the bureau of Registrations for your area or the hospital and have an amended because issued.