What do I need to do in order to get the father of my child to sign away his rights?

There is no next step. A parent can not just sign away his rights. If you are not married to him and do not depend on any public benefits for the child and neither one of you has done anything to establish his legal paternity, then nothing will happen. He does not get to see the child and you won't get child support. If you are getting benefits, then the state will go after him for child support so as not to burden the tax payer with the expense. However, he or you would have to take an active step for him to get to see the child. If there is someone who wants to adopt the child e.g. a new spouse, then rights can be terminated as part of the adoption process.

In WA. state, the only way a father can sign away his rights is if another is willing to adopt the child you cannot run away from being a parent. If that were allowed, no one would be paying child support and the Sate would have to support all of these kids whose father's want to avoid paying child support by signing away their rights. Its a common misconception.