Burgess began racing in 1950 with a Cooper Formula Three car and enjoyed a successful 1951 season that included a win at the Nürburgring. He subsequently moved into sports cars and Formula Two, where results were less impressive. His fortunes improved when he began a working relationship with Cooper Car Company, employed both in their factory and at the company's drivers' school based at Brands Hatch.
Racing a works Cooper Formula Two car in 1957, he finished fourth in the Oulton Park Gold Cup — a result that earned him a drive with Tommy Atkins' team in 1958 with a similar machine. That season he won at Crystal Palace and Snetterton, and took fourth places at Reims and Montlhery. A broken leg sustained in a crash at AVUS interrupted the year, but not before Burgess had made his Formula One World Championship debut for Cooper at the British Grand Prix.
Burgess returned to the World Championship in 1959 driving Scuderia Centro Sud's Cooper T51 with a Maserati straight-four engine. His best result of the season was sixth place at the German Grand Prix — the highest finish of his championship career. He continued with Scuderia Centro Sud into 1960 but managed only a tenth-place finish at the French Grand Prix from his starts, and was unable to qualify at Monaco.
For 1961 he joined the American Camoradi International team, making three starts from five entered Grands Prix. He used a Lotus 18 and later a Cooper T53, both Coventry Climax-powered, with a best finish of twelfth at the German Grand Prix. The following season saw him move to the Anglo American Equipe, where he drove a Cooper T59 in three championship rounds, finishing no higher than eleventh at the German Grand Prix.
His final Formula One season was 1963 with the Scirocco-Powell team, campaigning the Scirocco 02 fitted with a BRM V8 engine. Burgess retired from both the British and German Grands Prix, the latter being the last championship event of his career. Several other 1963 entries were withdrawn before the start.
Among his non-championship outings, Burgess was associated with the Aiden-Cooper chassis — a Cooper T59 modified by Tommy Atkins — at the IX International Gold Cup at Oulton Park in September 1962, a race that formed part of what one account described as a prelude to Paul Emery's final racing disaster. These non-championship events were a significant part of his racing programme, providing grid time beyond the limited championship outings his budget and team arrangements permitted.
Burgess represented the privateer spirit that characterised early 1960s Formula One, moving between a range of teams — Cooper, Scuderia Centro Sud, Camoradi, Anglo American Equipe, and Scirocco-Powell — in pursuit of competitive machinery. His career spanned the complete transition from front-engined to rear-engined Formula One cars. He died in Harrow, Greater London, on 19 May 2012, at the age of 81.