Rhodes drove a Cooper-B.M.C. Formula Junior car for the Midland Racing Partnership in 1961, winning on the Phoenix Park circuit in Dublin on 22 July and taking the Dunboyne Trophy on 29 July of the same year. He continued in Formula Junior through 1963, a season in which the Ford engine was by then required to be competitive. In June 1962 he drove Bob Gerard's Cooper-Ford in the 2,000 Guineas F1 race at Mallory Park.
Rhodes made his single World Championship start at the 1965 British Grand Prix at Silverstone on 10 July, driving a Cooper T60 Climax provided by Bob Gerard's team. He qualified on the 21st grid position and retired on lap 39 with ignition problems, not classified. He did not start any further World Championship events. He also appeared in non-championship Formula One events in 1962 and 1965 under Gerard's banner, including the Race of Champions and the BRDC International Trophy.
Rhodes competed in the BTCC from 1963 to 1969, primarily for the Cooper Car Company in Mini Cooper and Austin Mini Cooper S machinery. He opened his account in 1963, finishing 35th on 5 points with a ninth-place result in class. In 1964 he improved to ninth overall on 20 points, placing third in class. His strongest championship results came in 1965, 1966, 1967, and 1968, all of which he finished third or higher overall while winning the class title. In 1965 he placed third overall on 40 points and was class champion. In 1966 he rose to second overall on 50 points, again taking the class title. In 1967 he was again class champion, finishing third overall with 58 points. In 1968 he again won the class while finishing third overall on 50 points. He moved under the British Leyland Motor Corporation banner for 1969 and dropped to 18th overall with 18 points and 5th in class.
Rhodes produced his most notable sports car results in 1965 and 1966. At Le Mans in 1965, partnered with Paul Hawkins in a 1.3-litre Austin-Healey Sebring Sprite entered by the Donald Healey Motor Company, he completed 278 laps to finish twelfth overall and first in class — a well-regarded performance in a small-capacity car against larger machinery. At the 1966 Targa Florio, Rhodes partnered Timo Mäkinen in an M.G.B., finishing ninth overall and winning their class.
Rhodes is recorded as the 1968 European Touring Car Championship Division 2 champion, succeeding Andrea de Adamich and being succeeded by Spartaco Dini.