Kleive began to play drums in a rock and dance band, and then went to Oslo, where he studied at the Norwegian Academy of Music.

Described as one of the "leading lights in the Oslo Nu-jazz scene",[3] he made his distinctive contribution to groups like Terje Rypdal's Chasers,[4] Per Jørgensen's JøKleBa and Jon Balke's Magnetic North Orchestra as well as Marilyn Mazur & Future Song.

[2] Kleive has released the albums, Bitt (1996–97), Generator X (2000) and Ohmagoddabl (2004) on Bugge Wesseltoft's label, Jazzland.

[6][7] In 2013 he was engaged for a set on the Music Festival 'Punktfestivalen' in Kristiansand, where he also collaborated on recordings for the album Crime scenes (2006) with Jan Bang and Erik Honoré among others.

[8] With 1300 Oslo With Entra With Jokleba With Scheen Jazzorkester With Jon Balke With Kari Bremnes With Arve Henriksen With Jan Gunnar Hoff With Marilyn Mazur With Terje Rypdal With others