Named after a Polish grandfather, Teofil Joseph Kloska, who had settled in England, Kloska was brought up in Cornwall. As a child, he was taken to see a grisly outdoor production of Macbeth on Bodmin Moor, which made a great impression on him. After leaving school he attended University College London to read History and French, before training for an acting career at the Royal Academy of Dramatic Art, where he was in the same year group as Pip Carter, Kathy Rose O'Brien, Arthur Darvill, Sia Berkeley, Harry Hepple, Nathaniel Martello-White, and Danielle Ryan. He graduated in 2006.

In a Royal Shakespeare Company production of Imperium in 2017–2018, Kloska plays the slave-narrator Tiro, with one reviewer commenting on his "irrepressible wit and verve".