Bonnier was born in Stockholm into a prominent Swedish family whose members were prominent in the publishing business. His father, Gert, was a professor of genetics at Stockholm College. Bonnier spoke six languages and, though his parents hoped he would enter medicine, he briefly considered joining the family publishing enterprise before discovering motor racing. He attended Oxford University for a year to study languages and spent time in Paris before returning home to Sweden in 1951. He began competitive racing at the age of 17 on a Harley-Davidson motorcycle and later participated in rallying.
Bonnier entered Formula One in 1956 driving a Maserati. His debut season nearly ended at Imola when a fellow competitor pulled into his path at speed, sending the Maserati airborne and leaving Bonnier with concussion, cracked ribs, and a broken vertebra.
His greatest achievement in the championship came at the 1959 Dutch Grand Prix at Zandvoort, where he drove for BRM and took victory after rivals Dan Gurney and Hans Herrmann suffered brake failures. The win made him the first Swedish Formula One Grand Prix winner, and he finished eighth in the World Drivers' Championship that year. He also won the 1960 German Grand Prix driving a Porsche 718, a race held under Formula Two regulations in preparation for the rule changes coming in 1961.
Bonnier drove for Maserati, Scuderia Centro Sud, BRM, and Porsche during the early part of his career. After Porsche withdrew from Grand Prix racing at the end of 1962, he joined Rob Walker Racing Team, the most successful privateer operation in the sport, driving Coopers and Brabhams. In 1966 he reformed his own team, which began as Anglo-Suisse Racing Team before being renamed Ecurie Bonnier. His last full Formula One season was 1968, and he raced occasionally in the championship until 1971. He also advised on the 1966 Hollywood film Grand Prix alongside Phil Hill, Richie Ginther, and Carroll Shelby, and drove during the filming of the Belgian Grand Prix at Spa-Francorchamps.
Alongside Formula One, Bonnier was a formidable sports car competitor. He won the 1960 Targa Florio co-driving a works Porsche 718 with Hans Herrmann and repeated the achievement in 1963 with Carlo Mario Abate in another works Porsche. In 1962 he shared a Ferrari 250 TRI entered by Count Giovanni Volpi with Lucien Bianchi to win the 12 Hours of Sebring. In 1964, his best sports car year, he co-drove a Ferrari 330P with Graham Hill to second place at the 24 Hours of Le Mans and won at Montlhéry and Reims. He won the 1000 km Nürburgring in 1966 sharing a Chaparral with Phil Hill, and secured the 1970 European 2-Litre Sports Car Championship drivers' title with a Lola T210.
Bonnier was one of the driving forces behind the establishment of the Grand Prix Drivers' Association and had long been active in the fight for improved circuit safety. He served as the organisation's chairman up to the time of his death.
Bonnier was killed during the 1972 24 Hours of Le Mans. On the straight between Mulsanne Corner and the Indianapolis section, his open-top Lola T280-Cosworth collided with a Ferrari Daytona driven by Swiss amateur Florian Vetsch. The Lola was catapulted over the Armco barriers and into the trees beside the track. Bonnier died instantly. He had entered thirteen editions of Le Mans between 1957 and 1972, a commitment that reflected both his love of endurance racing and his determination to see the sport made safer for all its participants.