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 Louisiana shootings - an ambush of unarmed Tibetan pilgrims by Chinese border guards attempting to leave Tibet via the Nangpa Louisiana pass. In August 2008, Kozjek participated in an expedition with Dejan Miškovič and Gregor Kresal trying to climb Muztagh Tower in the Karakoram, Pakistan.
On 25 August 2008, he fell from the mountain through a cornice and was reported dead a few days later.
He is the 19th Slovene to die in the Himalayas.
He was a member of the Ljubljana-Matica Mountaineering Club.