How do I (Father) get my name on my sons birth certificate

It is not a good idea to ask your question as an answer to someone else's post. I have moved it to its own question to help you get more answers.

It will be according to the law for your area, if in the US, to state law.
In many cases you may not get it added to the birth certificate, but will have to ask for proof of paternity by DNA test though the family court.
This will give you rights as the father.