Kloska was born in North Cornwall and studied History at University College London and Paris-IV-Sorbonne before training as an actor at RADA. On graduation, he won the BBC Carleton Hobbs Radio Award.
Theatre highlights include four seasons with the Royal Shakespeare Company, culminating most recently in playing Leontes in The Winter's Tale for the first RSC/BBC Arts 'World Premiere' in the midst of the pandemic. Other work with the company includes Measure for Measure, Written on the Heart, The Christmas Truce and the critically-acclaimed role of Tiro in Robert Harris's Imperium in Stratford-upon-Avon and the West End.
Elsewhere on stage, he has worked at the Royal Court, the Manchester Royal Exchange, Sheffield Crucible, The Barbican, Regent's Park Theatre, Arcola Theatre, Orange Tree Theatre, Hampstead Theatre, Leeds Playhouse and at the Chichester Festival Theatre in Somerset Maugham's For Services Rendered directed by the late Howard Davies.
On television, Joseph has appeared as a series regular in Foyle's War (ITV), Pete Versus Life (C4) and played the role of Lord Porchester in the first two seasons of The Crown directed for Netflix by Ben Caron and Stephen Daldry. In film, he has worked with Mike Leigh (Happy-Go-Lucky and Peterloo), Kenneth Branagh (Cinderella), Nigel Cole (Made in Dagenham) and Cary Fukunaga (Jane Eyre). He has recorded over 100 plays and readings for BBC Radio and has recently narrated The Mirror and the Light by Hilary Mantel for Audible.
Alongside his acting career, Joseph is a director and theatre producer for the European Forum Alpbach in Austria.