If my deceased father and I are on the deed to a home that I have been the only one making the payments for at least 5 years, what rights do I have to this home?

If you and your father held title as joint tenants, then you survived to the property, and should obtain the assistance of probate counsel to assist you in obtaining title into your name; makes no difference whether you paid payments for the last five years, as those payments would be considered rent or a gift you made. As long as the payments are kept current, the bank probably will not ask you to refinance the property.