Bonnett was born in Hueytown, Alabama, a suburb of Birmingham, on July 30, 1946. He began his racing connection by working on Bobby Allison's cars and quickly became part of the Alabama Gang, which included Bobby, Donnie, Red Farmer, and later Davey Allison. He made his NASCAR debut in 1974 and recorded his first victory in 1977 at the Capital City 400 in Richmond, Virginia, driving for Harry Hyde–Jim Stacy Racing. That same year he added a second win at the Los Angeles Times 500, which would be the last Dodge victory in NASCAR until 2001.
In 1979, following a difficult period with the Hyde-Stacy operation plagued by financial trouble, Bonnett joined Wood Brothers Racing to replace the departing David Pearson. The move revived his career and he won three times in his first season with the team. He went on to win back-to-back World 600s at Charlotte Motor Speedway in 1982 and 1983, and back-to-back Busch Clash victories in 1983 and 1984 — four consecutive wins in NASCAR's premier non-points event.
In 1984 Bonnett moved to Junior Johnson's operation, becoming teammates with Darrell Waltrip. His best statistical season followed in 1985, when he finished fourth in the championship standings. He also participated in the International Race of Champions (IROC) in 1979, 1980, and 1984, finishing second twice.
Bonnett holds the distinction of winning the first NASCAR race held outside North America. In February 1988 he won the Goodyear NASCAR 500 at Calder Park Thunderdome in Melbourne, Australia, starting from pole and leading most of the race alongside Alabama Gang stablemate Bobby Allison. Cabin temperatures during the Australian summer event reportedly exceeded 57 °C inside the cars.
By the end of the 1980s Bonnett was still a consistent front-runner, and in the years that followed he showed flashes of his earlier pace before his career was derailed by injury.
On April 1, 1990, Bonnett was involved in a fourteen-car crash at the TranSouth 500 at Darlington. The collision with Sterling Marlin's car left Bonnett with amnesia and chronic dizziness that forced him into retirement from full-time racing. He transitioned into broadcasting, serving as a colour commentator for CBS Sports, TBS Sports, and TNN, and hosting the TNN programme Winners from 1991 until his death.
Bonnett nevertheless remained determined to race again. From 1992 he tested cars for friend Dale Earnhardt and owner Richard Childress. After receiving medical clearance in 1993, Childress gave him a ride in the DieHard 500 at Talladega on Earnhardt's recommendation. His comeback race ended with a dramatic crash in which the car became airborne before striking the spectator fence; Bonnett walked away unharmed and finished the race day calling the action from the CBS broadcast booth. He started the 1993 season finale at Atlanta but retired after three laps, the team withdrawing the car to help Earnhardt secure points in the championship battle.
For 1994, Bonnett secured a ride with James Finch's Phoenix Racing for at least five races, including the Daytona 500. On February 11, 1994, during the first practice session for the Daytona 500, a broken shock mount caused him to lose control of his Chevrolet on the high-banked fourth turn. The car swerved to the apron and then drove hard into the outside wall nearly head-on. Bonnett died from massive head injuries; he was 47 years old.
That same weekend, 1993 Goody's Dash champion Rodney Orr was also killed during Daytona practice. When Dale Earnhardt won the 1998 Daytona 500, he dedicated the victory in part to Bonnett. Bonnett is buried at Forest Grove Memorial Gardens in Pleasant Grove, Alabama. A road called Allison-Bonnett Memorial Drive in his hometown honours him alongside fellow Hueytown native Davey Allison, who had died seven months before Bonnett.
Bonnett ranked 47th in all-time NASCAR Cup victories at the time of his death. He appeared in two feature films during his career — the 1983 Burt Reynolds comedy Stroker Ace and the 1990 NASCAR drama Days of Thunder. He was inducted into the International Motorsports Hall of Fame and was portrayed by actor Sean Bridgers in the television film 3: The Dale Earnhardt Story.