Klink played with Glenn Miller from 1939 to 1942, heard on his weekly radio series for Chesterfield, and was one of the two tenor saxes on "In the Mood" (1939; the single was number one in the charts for 13 weeks). When Miller played fictional bandleaders in the films Sun Valley Serenade (1941, with Milton Berle) and Orchestra Wives (1942), Klink was among the band members who appeared (uncredited) on-camera
After Miller joined the military in 1942, Klink went on to play for Benny Goodman and Tommy Dorsey. Post-war, he began to establish himself as a session musician. He was also a member of the Sauter-Finegan Orchestra from 1952-1953. He also played with Louis Armstrong, Billie Holliday, Perry Como, Ella Fitzgerald, Frank Sinatra, and even Steve Allen on two albums. His only work as a bandleader himself was the 1955 Progressive Jazz, leading his own group (including drummer Ed Shaughnessy) for half of the dozen tracks (while trombonist Bob Alexander handled the rest).
In 1959, Jim Timmens (who would work with Klink again on the Sesame records) chose Klink as part of his group (variously billed as the "Jazz All-Stars" and "Swinging Brass Sound") on four jazz versions of established musicals. In the 1970s, he became a member of the World's Greatest Jazzband and continued to record until his retirement in the 1980s.