 His maternal great-grandfather was film-enthusiast E. T. Heron, who published "The Kinematograph Weekly".Dominic attended John Ball Primary School in Blackheath, London, followed by Thomas Tallis School in nearby Kidbrooke, then trained at the London Academy of Music and Dramatic Art (LAMDA) in acting and film editing, graduating in 2000.

Cooper 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", as the character Dakin, from its first reading.He also toured with the production to Broadway, Sydney, Wellington and Hong Kong as well as appearing in the radio and film adaptations of the play.