Cooper is from Greenwich, London. He attended the Thomas Tallis School in Kidbrooke, and trained at the London Academy of Music and Dramatic Art, graduating in 2000.

Cooper graduated from LAMDA in 2000. He first worked in television and film before making his stage debut in Mother Clap's Molly House at the National Theatre in 2001.

Cooper was involved in Alan Bennett's play The History Boys from its very first reading. He also appeared in the Broadway, Sydney, Wellington and Hong Kong productions and radio and film versions of the play. He has had notable roles in the Royal National Theatre's adaptation of the His Dark Materials trilogy where he played the lead character Will Parry, the TV series Down To Earth and Sense & Sensibility, and the film Starter for 10.

He appears in the forthcoming films The Duchess, An Education and Brief Interviews with Hideous Men and the BBC drama God on Trial. He is attached to star in the film City of Birds.

He also appeared on the front cover of Attitude magazine in July 2008.

Cooper was nominated for a 2006 Drama Desk Award for his work on the Broadway production of The History Boys.

He was nominated for the for Most Promising Newcomer (On Screen) award at the 2006 British Independent Film Awards and has been nominated for the British Film Critics Circle's 2006 award for British actor in support for his work on the film version of The History Boys.