As a student at North Kansas City High School, Conway excelled in football, track and field, basketball, and baseball.

Upon graduating, Conway was considered by many to be the best running back in football in the Kansas City metropolitan area and perhaps in the entire state as a senior and one of the best athletes of all time to hail from Missouri.

After high school, Conway was recruited to play football for coach Earl Blaik at the United States Military Academy.

After finishing college, Conway was a second round selection by the Philadelphia Eagles in the 1953 NFL draft,[2] but had his career ended early due to a string of injuries.

Conway had five children with his first wife Sue, who preceded him in death, Mike, Susie, Jim, Patty, and David.