Graham was born in Kirkby, Merseyside. He dropped his native Liverpudlian ("Scouse") accent for a London cockney accent in the Guy Ritchie film Snatch. He also adopted a Chicago accent when playing Baby Face Nelson in Public Enemies.
Following Snatch, Graham's highest-profile work in the US came in the HBO miniseries Band of Brothers and in Martin Scorsese's Gangs of New York. He is also featured in the music video for the Arctic Monkeys' number-one hit "When the Sun Goes Down", and the short film based on the song, Scummy Man, as well as having been featured in their video for the song "Fluorescent Adolescent". His performance in the Shane Meadows film This Is England was well-reviewed. While he plays the racist thug Combo in This Is England, Graham himself is part Jamaican by his paternal grandfather. He has also starred in the music video for Deadmau5 and Kaskade hit I Remember. He played Baby Face Nelson in Public Enemies, alongside Johnny Depp and Christian Bale. He donned a ginger wig to portray Billy Bremner, captain of the 1974 Leeds United football team, in the 2009 film The Damned United. He also grew a moustache to play Football Thug John Godden in Awaydays. He plays Scrum in Pirates of the Caribbean: On Stranger Tides.