Bergheim was raised in Asker and in Stabekk in the neighboring municipality Bærum, where he lived most of his life. He played in a number of bands and orchestras before devoting his professional life to music, in 1948. He was repeatedly named best Norwegian saxophonist when such contests were common in the 1950s and 1960s.
He led his own bands and orchestras starting in 1960, focusing mostly on swing.
Because of a lung ailment, he retired in 1990. Bergheim was for some time married to noted singer and actress Anita Thallaug.