Kuhnke raced motorcycles during the late 1940s before transitioning to car racing. He took up a Formula Three Cooper 500 and competed regularly through the 1950s, recording a number of wins and strong finishes. He also appeared in Formula Junior and Formula Two events before turning his attention to Formula One.
Kuhnke's first Formula One appearances came in 1962. He borrowed Wolfgang Seidel's Lotus 18 — initially fitted with a Climax engine and later a Borgward unit — and entered two non-championship races. He failed to qualify for the Pau Grand Prix and retired from the Solitude Grand Prix with engine failure.
He subsequently prepared a Borgward-engined Lotus of his own for use in four further events planned for the second half of 1962. The Borgward unit was an old sports car engine with twin camshafts and direct fuel injection, an unusual choice that required substantial preparation work. He missed all four scheduled events due to ongoing problems with the engine preparation.
In 1963 Kuhnke campaigned what became known as the BKL Lotus, a designation formed from the initials of Borgward, Kuhnke, and Lotus. The car was a standard Lotus 18 chassis modified by Kuhnke himself, retaining the Borgward engine.
In non-championship events he failed to qualify at the Rome Grand Prix alongside team-mate Ernst Maring, and both cars retired from the Solitude Grand Prix in July with engine failures.
Kuhnke's only World Championship Formula One entry came at the 1963 German Grand Prix at the Nürburgring, where his car carried the number 27. He failed to qualify by a considerable margin. Later that year he retired from the Kanonloppet at Karlskoga Circuit in Sweden with fuel injection problems, after which he stepped back from driving.
Kuhnke continued in motorsport as an entrant after retiring from driving. At the 1964 Solitude Grand Prix he fielded two BKL Lotus cars for Maring and Joachim Diel, a German helicopter pilot. Diel's car was among seven eliminated in wet-weather accidents on the opening lap. Maring survived to finish tenth and last, four laps down on the leader — the only time a Kuhnke-entered car reached the finish line of a Formula One race.
Kuhnke represents a type of post-war German motorsport privateer: technically ambitious, self-sufficient, and working at the outer margins of Formula One with home-built machinery and unconventional engine choices. The BKL Lotus, with its Borgward engine unit transplanted from a sports car application, was an idiosyncratic creation whose fuel injection system proved more problematic than competitive. Kuhnke died in Braunschweig in February 1969 at the age of 58, several years after his last involvement in Formula One competition.