What is the process of a father signing over his rights?

It's not something you just "do". Very few courts will allow a parent to walk away from their responsibilities. Usually there is a history of neglect and/or abuse, where the court is already involved. Regardless of the circumstances, the child is the top priority. You might be able to relinquish custodial rights, but not financial obligations. Be that as it may, you can speak to the Family Court Clerk in your jusrisdiction and he/she will inform you of the process for the state you live in.Don't expect it to be a pleasant experience though.