Hawkins was the son of a racing motorcyclist who later became a church minister. He began his racing career in Australia in 1958 with an Austin-Healey. In 1960, he moved to England and found employment with the Donald Healey Motor Company Ltd. under John Sprinzel, who described Hawkins as a "really good guy" and a "very good mechanic."
Hawkins quickly began racing for Donald Healey, driving an Austin-Healey Sprite. He won his class in a GT race at the Aintree 200 meeting on 30 April 1960. Later that year, he finished 38th at the Nürburgring 1000 km race with co-driver Cyril Simson. In 1961, he teamed with John Colgate at Le Mans in an Austin-Healey Sprite, but they retired due to engine problems. On Whit Monday, 1962, Hawkins drove Ian Walker's Lotus-Ford to victory in the up to 1,150 c.c. sports car race at Crystal Palace, setting lap and race records. In 1965, at Le Mans, Hawkins and John Rhodes finished twelfth overall and first in class in a 1.3-litre Austin-Healey Sebring Sprite entered by the Donald Healey Motor Company, completing 278 laps.
Hawkins also competed in single-seaters. He participated in the first race run under the new Formula Two regulations at Pau on 5 April 1964, finishing seventh in an Alexis. He won the Formula Two Eifelrennen race on the Nürburgring south circuit in an Alexis-Cosworth Mk. 7 on 25 April 1965, in challenging weather conditions.
Hawkins participated in three Formula One World Championship Grands Prix, making his debut on 1 January 1965 at the South African Grand Prix in a Brabham Formula Two car with a pushrod Ford 1500cc engine. He started with the John Willment Automobiles team, like fellow Australian Frank Gardner. His best finish was ninth on his debut, and he did not score any championship points.
He achieved two podium finishes in non-championship Formula One races in South Africa, finishing second in the 1964 Rand Grand Prix and third in 1965. In 1964, he won the Rhodesian Grand Prix in the Brabham, and in 1965, he also won the Cape South Easter Formula One Trophy. He was a non-starter in the 1965 British Grand Prix and retired from the German Grand Prix that year due to an oil leak.
Hawkins is one of only two Formula One drivers, along with Alberto Ascari, to have crashed into the harbor during the Monaco Grand Prix. This occurred during the 1965 race, when he spun at the chicane after 79 of 100 laps. He was unhurt in the incident, with reports stating he "struck the wooden barrier at the entry and spun through the straw bales and over the edge of the quay and into the harbour. The Lotus sank to the bottom and the rugged Australian bobbed to the surface and struck out for shore, while boats went to his rescue."
Hawkins achieved significant success in the World Sports Car Championship. On 14 May 1967, he won the Targa Florio in Sicily, co-driving a factory-entered 8-cylinder Porsche 910 with Rolf Stommelen. On 23 May 1967, he finished second in the Nürburgring 1000 km in a Porsche 910. He also won the Zeltweg 500 km race on 20 August 1967, driving a Ford GT40. Later that season, on 15 October 1967, Hawkins and Jacky Ickx won the Paris 1000 km race at Montlhéry in a J.W. Automotive Mirage.
On 25 April 1968, Hawkins won the Monza 1000 km race with David Hobbs in a Ford GT40. He finished second in the Watkins Glen 6-hour race, again with Hobbs, and secured third-place finishes at the Nürburgring 1000 km with Jacky Ickx and the Zeltweg 500 km races. On 23 November 1968, he won the Cape Town Three Hours solo in a Ferrari P4.
In the spring of 1969, Hawkins moved his racing shop from North London to Slough as he was developing a business as an owner/operator of racing cars. He was killed on 26 May 1969, when his Lola T70 MkIIIB GT crashed and burned at Island Bend during the RAC Tourist Trophy at Oulton Park. Contemporary Mike Hailwood commented on Hawkins' death in the Daily Mirror, stating, "The news of his death horrified me. I can hardly believe that a man as skilful and as experienced as he was should be killed in this way."