McGriff grew up in rural Houston County, Alabama in the southeastern corner of the state. He attended and played football at Cottonwood High School in Cottonwood, Alabama graduating in 1976. He played college football at the University of Alabama in Tuscaloosa, Alabama for the legendary coach Paul "Bear" Bryant from 1976 to 1979. McGriff was an interior defensive lineman for the 1978 squad that defeated Penn State in the famous “goal line stand” Sugar Bowl. He was also part of the 1979 team that repeated as undisputed national champions, punctuated by a win over Arkansas in the Sugar Bowl.
McGriff entered the NFL and signed with the New York Giants in 1980 as a defensive lineman. He made the NFL's all-rookie team that year and was a member of the Giants' 1986-1987 Super Bowl Championship team. McGriff was primarily used as a defensive tackle and rotated into the game on downs when a running play was anticipated (McGriff would be rotated out of the game and a pass-rush specialist would be brought into the game on passing downs.) He played his final season (1987) with the Washington Redskins.