Klingberg was born in Gothenburg, Sweden, to Anders Klingberg. His father, Anders, is also active in hockey and was a head coach for Göteborg during TV-pucken, a Swedish ice hockey tournament. Klingberg also has two brothers, both of whom are active hockey players; Carl who currently plays for EV Zug of the National League (NL) and Olle, who currently plays with Lerums BK.
