Can my wife add my last name to the end of her child's last name, he has his father's last name,just wondering if mine can go after biological father's last name?

He can go by any name you want him to use, but legally, as long as he is a minor, he still has to have his biological father's last name unless BOTH parents agree to the change. At least that's the law in Ohio.

Legally a person can use any name they want as long as there is no intent to defraud. However, he will still have to use his legal name on any legal papers. And his legal name is what appears on the birth certificate.

To change that you will need to adopt him or he will have to wait to he is 18 and can petition a court for a name change.