Leslie was a five-time Scottish karting champion before moving to single-seater cars, winning the Formula Ford title in 1978. He raced in Formula Atlantic in 1980 before spending four years in the British Formula Three Championship from 1981 to 1984, becoming associated with the Ecurie Ecosse team.
Through Ecurie Ecosse, Leslie moved into the World Sportscar Championship in the mid-1980s, competing in the C2 class. The team earned the 1986 C2 class championship, and Leslie finished second in the Drivers Championship in 1987 alongside teammate Ray Mallock. He and the team also finished second in the C2 class and eighth overall at the 1987 24 Hours of Le Mans. He went on to contest Le Mans a total of ten times throughout his career, including driving a works Jaguar XJ220 in 1993 alongside Win Percy and Armin Hahne, retiring with engine failure while in contention for the GT class lead. In 1990, following Ecosse's Aston Martin programme transition, Leslie departed to join Tom Walkinshaw's Jaguar sportscar team.
Leslie maintained his links with Ecurie Ecosse when the team entered the British Touring Car Championship in 1990, competing on a partial schedule for two seasons before becoming a full-time Vauxhall driver in 1992. He took his first BTCC win in 1993 and accumulated six pole positions across those two seasons with consistent top-ten championship placements.
An unsuccessful season in a Mazda in 1994 preceded a move to Honda for 1995, when the Japanese manufacturer entered the BTCC for the first time. Despite the car arriving late to pre-season testing with early reliability problems, Leslie finished tenth overall after a strong end to the season. In 1996, following a difficult start with several collisions, a victory in the British Grand Prix support meeting launched a strong second half that resulted in fourth in the championship.
For 1997, Leslie switched to Nissan alongside Anthony Reid while James Thompson and Gabriele Tarquini took the Honda seats. The Nissan Primera was initially uncompetitive, but much of the credit for its subsequent improvement was attributed to Leslie's development expertise. In 1998 he converted that progress into two victories and five podiums, finishing seventh in the standings. He also made a one-off British GT Championship appearance at the British Grand Prix supporting race, finishing third in a Porsche 911 GT1 shared with Matt Neal.
Leslie's finest season came in 1999, when he finished as runner-up in the BTCC to his Nissan teammate Laurent Aïello. He took three race wins and ten podiums across the campaign, but the consistency of Aïello ultimately proved decisive. After Nissan withdrew at the end of the season, Leslie made only occasional BTCC appearances in 2000 while also contesting rounds of the European Super Touring Cup and the Swedish Touring Car Championship.
Leslie joined Proton for 2002 and 2003 in the BTCC but the programme had limited resources and the best results were a pair of third-place finishes. After leaving competitive BTCC racing, he took up commentary work for Eurosport on the World Touring Car Championship and later lectured in Motorsports Management at Swansea Metropolitan University. Allan McNish credited Leslie with giving him, David Coulthard, and Dario Franchitti crucial early career support.
Leslie died on 30 March 2008 when a private jet he was travelling in crashed into a housing estate in Farnborough, Hampshire. He was travelling with racing driver and team owner Richard Lloyd and data engineer Chris Allarton to the Nogaro Circuit in France for a test session ahead of the FIA GT3 European Championship. The investigation found that incorrectly identified non-critical system failures led to the in-flight shutdown of both engines, which the crew were unable to restart before impact.