Hart was raised just outside of Chicago for the first few years of his life, until his father died when he was seven. His mother remarried, and Hart's stepfather inspired him to play sports, even pushing him out of a car to participate in a football skills competition, which he won. It was at Niles West High School in Morton Grove at age 13 when he started playing football as a quarterback, though he lettered in basketball for three years and also played baseball. He received a football scholarship to play for the Southern Illinois Salukis from 1963 through 1965. After not being drafted in the 1966 Draft, Hart's former coach Don Shroyer invited him to a tryout with the Cardinals. He impressed the team and was signed soon after.
After playing in just one game in 1966, current quarterback Charley Johnson left to fulfill an ROTC commitment, leaving Hart to be the starter, and while his teams early in his career were not particularly great (31–33–5), his and the teams' fortunes increased with the hire of coach Don Coryell in 1973. From 1974 to 1976, he guided the Cardinals to three straight ten-plus-win seasons along with back-to-back division crowns in 1974 and 1975, leading the Cardiac Cards to ten game-winning drives during that three-year span. However, he went a combined 40 for 81 in passing with two touchdowns and 4 interceptions as his Cards lost twice. By the time the Cardinals returned to the playoffs again at the conclusion of the 1982-83 season, Neil Lomax had already established himself as the starter and Hart did not play. He was released by the team after that season. Hart was signed by the Redskins to back up Joe Theismann for the 1984 season, playing little in that season before retiring in the off-season. He was also selected to the Pro Bowl four times. In the 1977 Pro Bowl, Hart threw five interceptions, the most in the Pro Bowl's history. He went 87–88–5 in his career, was sacked 243 times, and played in 201 games. He is 25th in passing yards, 29th in victories, 34th in completions, and 32nd in passing touchdowns, though he is 10th in passes intercepted (once having a 30-interception season in 1967), 73rd in being sacked, and 161st in Passer rating. 
In 1983, Hart and fellow teammate Dan Dierdorf opened up Dierdorf and Hart Steak House. Hart broadcast games on WGN with Dick Butkus after his retirement, served until 1989, quit to focus on being the athletic director for Southern Illinois University Carbondale, and served until a chancellor changeover forced him out in 1999. Hart has been married to his college sweetheart for over 40 years; he has three children and four grandchildren. He resides in Naples, Florida, and participates in golf tournaments organized by former players.