Björn Kleiman holds a BMus Hons from the Royal College of Music in London, where his teachers included Maureen Smith and Dona Lee Croft. In addition to regular lessons, Björn took part in master classes with Lewis Kaplan, Bela Kantona, Hugh Bean and Benjamin Schmid. After graduating from the RCM he continued his studies at the Hochschule für Musik in Karlsruhe, Germany, with Professor Ulf Hoelscher, and his assistant Karl-Heinz Schultz.

Björn has given concerts in Sweden, England, France, Germany and Switzerland and has appeared as soloist with orchestra in concertos by e. g. Bruch and Wieniawski. A strong commitment to contemporary music has resulted in him collaborating with the most prominent composers in Sweden today. He has premièred pieces by Albert Schnelzer and Stefan Klaverdal and has given UK premiers of works by e. g. Erland von Koch and Jesper Nordin. Many composers have also written works especially for Björn i.e. ”The Longing of Eurydice” by Stefan Klaverdal, commissioned by the Swedish Concert Institute and first performed in Paris in 2006. In 2007, Björn Kleiman's CD entitled "Dr Caligari" was released on the label C-Y contemporary. The CD comprises solo works as well as compositions for violin in combination with piano and computer by Schnelzer, Staern, Larsson-Gothe and Klaverdal. The disc received rave reviews and has been broadcasted on BBC Radio 3, Latvian Classic Radio, Swedish Radio P2 as well as in the USA.

Apart from his work with Kleiman Klöfver Duo, Björn performs regularly with pianists Martin Sturfält and Simon Lepper. Together with Martin Sturfält, Björn has signed with the German label Thorofon to record the complete works for violin and piano by Harald Genzmer. The disc is due for release in the spring 2010.