[1] Kjellström was born in Stockholm and started to play ice hockey and football with local clubs Eriksdals IF and Helgalundsklubben as a youngster.

Kjellström had his major breakthrough scoring a hat-trick when Hammarby won 3–1 against German giants SC Riessersee in an exhibition game on 29 January 1939.

Ahead of the 1939–40 season, Kjellström joined fierce rivals AIK, but decided to leave the club after one year to return to Hammarby IF.

[5][1] Due to World War II, Kjellström only played 15 games with the Sweden national team, scoring a total of 21 goals.

[1] After playing with the youth teams of Hammarby IF, Kjellström started also chose to represent AIK in football when he switched clubs in ice hockey.

Subsequently, Kjellström made his debut in Allsvenskan, the domestic top tier, in a 2–0 home win against IK Sleipner on 22 October 1939, aged 18.