Norris is from Romford, Essex, England. He attended Redden Court School and then went to the Central School of Speech and Drama in London, graduating in 2008, winning the prestigious Alan Bates Bursary.
Norris started his career on stage and has continued to. His work on stage include A Puzzle at the Royal Court, Hamlet at the Royal Shakespeare Company and Antigone at the National Theatre. He also starred in A View from the Bridge at the Young Vic, which was critically acclaimed.
Norris has also wrote plays including Goodbye to All That, performed at the Royal Court, Hearts at the National Theatre Connections and the Shed Theatre and Growth at Paines Plough’s Roundabout Theatre which opened to rave reviews. His works Goodbye to All That and So Here We Are were performed to critical acclaim at the Royal Exchange.