What process or forms do I need if a parent wants to give up his rights?

The father is unable to give up his rights just because he wishes to do so.  This could be done in the event that you marry and your spouse wants to file a 'step-parent adoption' and at that point the biological dad would give up his rights.  In effect, if the child was adopted then rights would be given up. The father does not need to visit the child or be part of the child's life.  However, if you want child support you could request it.