After graduating from acting school, Kjellson was hired by the royal theater Dramaten in 1946, and would perform 40 roles during his five years there. In subsequent years, he worked at Upsala city theater (1951-1957) and Stockholm city theater (1960-1964). In 1964, he would return to Dramaten and has remained there ever since, as both an actor and a director. The 2005-2006 season would mark his 140th role in the theater. His stage credits include Lucky Pehr, The Three Musketeers, Richard III and Macbeth.
Kjellson was also in of the first generation of TV theater actors, where he played 30 roles from 1958 to 1962. He received his first minor movie role in Blåjackor (1945).
He is married to the Swedish actress Meta Velander.