My child's father is not in the picture and has not been for six months. He has only seen her a handful of times. He is not currently paying child support. What is the law about changing her last name to mine. She currently has her father's last name.

Is he on the birth certificate? If so, you need to go to court to get a legal name change. However, you can use any name you want as long as there is no intent to defraud. So you can just use your names on any papers you need to fill out.