Köhler began training for the luge in programs sponsored by the East German government. He won his first world championship luge title in 1962, competing in the singles event. In 1967 he again won the singles title, and that year he also added the two-seater title, sharing the luge with Klaus Bonsack.

The sport of luge tobogganing debuted as a medal competition at the Winter Olympics of 1964, and Köhler became the first luge Olympic gold medalist by winning the singles event. He competed again in the Winter Olympics of 1968 at Grenoble, France. For the first time at a Winter Games, there were two separate German teams, and controversy swirled around the East German teams from the start. After the East German women's team was disqualified for heating their runners, other teams declared that they would walk out if the men's team was allowed to compete. The International Luge Federation chose not to suspend the men's team; however, mild weather prevented further runs on the luge course, and the competition ended after three runs. Despite these obstacles, Köhler was able to win another gold medal in the men's two-seater (again with Bonsack) and a silver in the singles event.