My fiancé wants to adopt my kids and the natural father wants to relinquish his rights. Is this possible. I am aware he will not be paying child support any longer. How can this be done.

Yes its possible, but I would wait until after you are married. The courts would prefer to see the commitment before allowing the adoption.

You will need an attorney to draw up the papers, which would include the relinquishment of rights for the bio father. I would shop around for an attorney now since some states have a waiting period after the wedding before they will process the adoption.