How does my husband go about filling an adoption if my sons biological father is not on the birth certificate?

You should consult with an experienced termination/adoption attorney in your area to discuss your matter. Generally speaking, unless you can obtain a consent to the adoption by the biological father you will have to file a termination of parental rights action. If the biological father is not on the birth certificate then he might have to submit to a DNA test to confirm paternity. You can see why it is essential to speak with a local and experienced attorney to have your questions answered.

File a Step Parent Adoption proceeding.

The best way to go about an adoption in your situation is to find a good (not just any) lawyer who is an expert in adoption law. Adoption, if done improperly, can be a disaster, especially in a situation such as yours, where the father's whereabouts are unknown. I could tell you the basic steps for adoption, but they'd be meaningless if you don't get a lawyer to help you. It doesn't have to be me you hire, but hire a good adoption lawyer. It's well worth the investment to see that it's done right.