Kozjek was born in Setnica near Polhov Gradec, SR Slovenia, Yugoslavia. He was a member of the Ljubljana Matica Alpine Club. In 1997, he was the first Slovene climber to ascend Mount Everest without supplemental oxygen. In October 2006, he led a new route up the southeastern face of Cho Oyu in less than 15 hours and took photographs of Nangpa La shootings - an ambush of unarmed Tibetan pilgrims by Chinese border guards attempting to leave Tibet via the Nangpa La pass. For his accomplishments he received the Piolet d'Or 2006 Spectators Choice.