Leslie was Scottish karting champion 5 times before switching to cars, winning the Formula Ford title in 1978. He later moved to the British Formula Three Championship from 1981 to 1984, becoming involved with the Ecurie Ecosse team. With Ecosse, he moved to the World Sportscar Championship, driving to multiple C2 class victories and helping the team earn the 1986 championship. Leslie himself would earn second place in the Drivers Championship in 1987 alongside teammate Ray Mallock. Leslie and the team also finished second in the C2 class and 8th overall in the 1987 24 Hours of Le Mans. Ecosse eventually took over the Aston Martin sports car program before Leslie departed in 1990 to become part of Tom Walkinshaw's Jaguar team. Leslie competed at the 24 Hours of Le Mans a total of 10 times, including driving the works Jaguar XJ220 in 1993 alongside Win Percy and Armin Hahne, only to retire with engine failure while in contention for the lead of the GT class.