How do I go about asking my ex to sign his rights off his child?

I order for your "ex" to sign off on his parental rights is to sign a document stating such where he will no longer be liable for any child support for the child where in return he gets no visitiation or contact. The agreement needs to be approved by a court order. The problem that you may have from a public policy standpoint is that unless someone takes on his parental rights other than yourself, the request that you seek may not be approved by the court.

The rationale is that the states in this country have a vested interest that unless one parent passes, they both are obligated for child support payments of a minor child they both have.

Your first step is to see if he's even willing to waive his rights to the child.  If he has no interest in the child and has had no involvement, he may simply agree to sign an order terminating his parental rights.  You would need to have a family law attorney draft you a proposed set of papers and see if he'll sign them.  If he refuses to sign the paperwork, then you may have a hard time getting the rights terminated since he has been paying child support as required.