Acheson grew up in Cookstown. His father, who owned an Ulster brickworks, competed in motorcycles and Formula Ford in the 1970s. In 1976, Acheson tested his father's old Crosslé FF1600 at the local Kirkistown Circuit and was lapping within three seconds of the lap record on his first visit. He entered his first race at Kirkistown that year and, after his father agreed to purchase new machinery on condition that Kenny quit smoking, Acheson went on to win the 1977 Northern Ireland FF1600 Championship.
He moved to England in 1978 to contest three Formula Ford series, driving Royale machinery with RMC sponsorship for Alan Cornock. He won 29 races and all three championships, and received a Grovewood Award at year's end. In 1979 he stepped up to British Formula Three and set the fastest lap at the F3 support race for the British Grand Prix, though he took no championship wins. The 1980 season saw him lead the Vandervell British F3 Championship before Stefan Johansson — equipped with a new Ralt RT3 — closed the gap and took the title at the final round.
In 1981 Acheson joined Docking Spitzley Racing in Formula Two with a Toleman TG280. A serious accident at the Pau street circuit, where a wheel-to-wheel battle with Michele Alboreto left him with both legs badly broken, ended his season prematurely. He returned in 1982 with Ralt Honda, finishing seventh in the European Championship. In 1983 he drove RAM March F1 cars in seven races, failing to qualify on six occasions before finally making the grid at the South African Grand Prix, where he finished twelfth. In 1985 he returned to RAM Racing as a substitute for Manfred Winkelhock, who had been killed in a sportscar race at Mosport Park. With the more competitive Hart-powered RAM 03, Acheson qualified for both the Austrian and Italian Grands Prix before funding dried up.
After a brief and unsuccessful foray into CART — entered for the 1985 Indy 500 but did not drive, then crashing or failing to qualify at Meadowlands, Elkhart Lake, and Laguna Seca — Acheson relocated to Japan. He raced for Kunimitsu Takahashi's Advan-backed Alpha team in Formula 3000 and sports cars, winning the Japanese Sportscar Championship in 1987.
That success earned him a place with Sauber-Mercedes for 1988 and 1989. He was scheduled to race at Le Mans in 1988, but the team withdrew during practice. The 1989 season proved more fruitful: Acheson completed a full campaign with Sauber-Mercedes, finishing second at Le Mans with co-driver Mauro Baldi, and going on to win at Brands Hatch and Spa with Baldi.
Despite strong performances, Acheson was dropped by Sauber at the end of 1989. He moved to Nissan for the 1990 World Sportscar Championship season, but retired the R90CK from Le Mans with a gearbox failure on the warm-up lap. In 1991 he drove for Silk Cut Jaguar at Le Mans and finished third in the XJR12. A return in 1992 with Toyota in the Tony Southgate-designed TS010 yielded another second place, before a failure to finish in 1993. One final Le Mans appearance came in 1995, driving the SARD MC8R; total brake failure after just 14 laps ended the race early.
In 1996 Acheson drove a Newcastle United-liveried Lister Storm at the Daytona 24 Hours alongside Geoff Lees and Tiff Needell. In the closing moments of the race the Lister was destroyed in a violent crash when a slower car hit him. He walked away uninjured from the wreck and subsequently walked away from professional racing altogether.
Acheson's career is that of a driver who reached Formula One through talent and tenacity but found his greatest consistency at Le Mans and in Japanese sports car racing. His repeat appearances at Le Mans across multiple eras — Sauber-Mercedes, Nissan, Jaguar, Toyota, SARD — across eight years (1989–1995) illustrate both his versatility and his value as a steady long-distance hand. The 1989 Sauber-Mercedes season, with a Le Mans second and victories at Brands Hatch and Spa, remains the high-water mark of a pragmatic career shaped as much by funding realities as by outright speed.