Rensen was born in Liverpool. His father was Dutch and worked as an executive with a firm in Crosby, where Ralph attended St Mary's College. He is buried in the Borough Cemetery in Douglas, Isle of Man, located almost directly opposite the grandstand.
Rensen made his Isle of Man debut in the 1953 Manx Grand Prix in the 350cc Junior class, though he did not finish. In the Senior class that year he completed the race in 33rd position. The following year he moved up to 14th in the 350cc category, and from 1955 onward he competed annually in the Junior TT, the Senior TT, and the Ulster Grand Prix.
Throughout this period Rensen was predominantly a Norton Manx rider. Occasional departures included a 250cc Velocette in 1957, which he failed to finish, and a one-off appearance on the NSU Sportmax belonging to Fron Purslow in 1960, after Purslow had been injured in training. In 1959 he also started in the 500cc class at the French Grand Prix.
His best results arrived in 1960 at the Ulster Grand Prix, where he finished fifth in the 350cc class and fourth in the 500cc class after a photo finish. These performances placed him 12th in the final world championship standings in both the 350cc and 500cc categories — his first points in either class. During the 1960 Cookstown 100, Rensen publicly stated his intention to stop racing following the fatal crash of his friend Dave Chadwick at Mettet.
Despite his stated intention to retire, Rensen returned to racing in 1961 after being contracted by Bultaco as a factory rider. His season opened at the German Grand Prix on the Hockenheimring, where he failed to finish with his Norton in the 500cc race but placed fourth in the 350cc category.
At the Isle of Man TT, he delivered strong results: third in the Junior TT behind Phil Read and Gary Hocking, and sixth in the Lightweight 125cc TT on his Bultaco. During the Senior TT, Rensen crashed fatally near milestone eleven of the Snaefell Mountain Course on 16 June 1961. He was 28 years old. Posthumously, he was classified sixth in the 350cc World Championship and 19th in the 125cc class for that season.
Ralph Rensen competed at the Isle of Man TT in every year from 1953 to 1961, a record of commitment that reflected the culture of British club racing in that era. His career arc — from backmarker to factory signing — speaks to steady, determined progress; his death on the Mountain Course placed him among the many riders of that generation who gave their lives to the sport's most demanding venue.