McGraw was born in Vinita, Oklahoma. He is the son of Jerry Stevens and Joe McGraw. He grew up with two older sisters, Deana and Donna, and younger sister, Brenda. They lived in the oil fields of North Texas. His father was an equipment supplier there. During McGraw's childhood, his family moved so his father could become a psychologist. McGraw attended Shawnee Mission North High School in Mission, Kansas. In 1968, he was awarded a football scholarship to the University of Tulsa. He played middle linebacker. His coach was Glenn Dobbs. Glenn Dobbs is the father of Gary Dobbs. Gary Dobbs went into business with McGraw. On November 23, 1968, McGraw's team lost to the University of Houston 100-6. This is one of the most badly played games in college football history.