Graham began racing at Liverpool's Stanley Speedway on dirt. His first road race appearance came in 1929 on the Oswestry Park Hall circuit, where he finished second riding a secondhand Dot-JAP. Through the early 1930s he campaigned a series of Rudge hybrids with modest success.
In 1936 he acquired a 250cc OHC OK-Supreme and entered the Ulster Grand Prix, though mechanical failure ended his race. After rebuilding the engine, he won at Donington Park before finishing fourth at the Ulster Grand Prix the same year. His results brought him to the attention of John Humphries, son of OK-Supreme's founder, who offered him a position assembling OHC engines. Graham, Andy McKay, and Humphries formed a trio of OK-JAP riders known in Midlands racing circles. He finished 12th in the 1938 TT Lightweight class and was preparing to ride a Velocette for 1940 when the outbreak of war intervened.
Graham served in the RAF from 1940, assigned to 166 Squadron flying Lancaster bombers over Germany. He attained the rank of Flight Lieutenant and was awarded the Distinguished Flying Cross in December 1944. He subsequently flew with Transport Command before being demobilised in 1946. On returning to civilian life, he joined Associated Motorcycles through an invitation from Wing Commander Jock West.
Graham resumed racing as a member of the AJS factory team, winning at the first post-war Cadwell Park meeting on a Norton 350. In 1947 he placed ninth in the Senior TT on an AJS Porcupine. At Montlhery in 1948, Graham, West, and French rider Georges Monneret broke 18 world records at speeds between 107 and 111 mph.
The inaugural Motorcycle World Championships were held in 1949. Graham entered on the AJS Porcupine and led the opening round at the Isle of Man TT by 90 seconds before a magneto failure saw him push home in ninth. He won Round 2 at Bremgarten in Switzerland and set the fastest lap. He finished second in the Dutch TT behind Nello Pagani, retired at Spa, then won the Ulster Grand Prix and again recorded the fastest lap. At the final round at Monza, Pagani won for Gilera, but with scoring based on a rider's best three finishes, Graham's two wins and a second narrowly outweighed Pagani's two wins and a third. Graham became the world's first 500cc motorcycle world champion.
In 1950, Graham finished third in the championship behind Umberto Masetti and Geoff Duke. In 1951, Count Domenico Agusta approached him to ride and develop MV Agusta's 500cc four-cylinder machines. While the MV was powerful, its handling was difficult to manage. Graham failed to score in the 500cc class for MV that year, but rode a Velocette in the 350cc class and won the Swiss Grand Prix.
In 1952 he delivered MV Agusta's first-ever 500cc Grand Prix victory at Monza, followed by a second win in Spain. Despite these results he finished the season second in the championship to Masetti. He also placed third in the 250cc class and fourth in the 125cc class that year.
For 1953 Graham was considered the pre-season favourite. He won the prestigious Mettet Grand Prix invitational and then achieved his first TT victory in the 1953 Lightweight 125cc class for MV Agusta. However, during Friday's Senior TT, he lost control at high speed as his machine crested the rise after the bottom of Bray Hill. The forks jammed on full compression, and Graham was killed instantly. He was 41 years old.
Leslie Graham was the first rider ever crowned world motorcycle champion, and his title in 1949 anchors the sport's entire historical record. A commemorative shelter, the Graham Memorial, was erected on the Snaefell mountain road in 1955 in his honour. His wartime service record and post-war achievements made him one of the most admired figures in early British motorsport.