How do I add the name of the father to the birth certificate?

Adding the name of the father to the birth certificate requires filing an Acknowledgment of Paternity. Acknowledgement of Paternity forms can be found at all birthing hospitals, the Office of Vital Records, the Division of Child Support Services at the Arizona Department of Economic Security offices and many of the office of the county registrar. You can also work with the courts to file for voluntary paternity or file for paternity, legal decision-making authority and related matters together. Your court may have a do-it-yourself packet. If not, many courts accept the packet of Maricopa County.
Resources:
•    Arizona Department of Health Services Vital Records – Corrections & Amendments to Birth Certificates
•    Division of Child Support Services – Voluntary Paternity Program
•    Maricopa County  To Establish Voluntary Acknowledgement