Graham was born in the Southdene district of Kirkby in Lancashire, a satellite town of Liverpool, to a paediatric nurse mother and a social worker father. His paternal grandfather was Jamaican and one of his grandmothers was Swedish. He has portrayed characters from several British, Irish and American regions, including Cockney, a Geordie, a Liverpudlian, and a Scotsman; an Irish gang member in 19th-century New York City, a 1930s Chicago bank robber, a Brooklyn mobster, a Texan, and the short-fused English nationalist, "Combo", in This Is England. He was nominated for an RTS Award for his work in The Street and a British Independent Film Award for his work on This Is England.

He appeared in the music videos for "I Remember" by Deadmau5 and Kaskade, "Fluorescent Adolescent" and "When the Sun Goes Down" by the Arctic Monkeys and "Unlovable" by Babybird, directed by Johnny Depp. Graham has also appeared in the music video "Soul Vampire" by Macclesfield-based neo-psychedelic band The Virginmarys, and the music video for "Turn" by Travis, also Kasabian's video "You're in Love with a Psycho". He appeared as an inmate within the open prison system for Goldie's 'I Adore You' music video in Feb 2017. In 2011, Graham starred in the BBC Christmas show Lapland, but when the series was announced, called Being Eileen, Graham's role was recast to Dean Andrews. He also appeared on a promotional video for a DLC of the video game Call of Duty: Ghosts, titled "CODnapped". Graham is a fan of Premier League football team Liverpool F.C. and has appeared several times on Sky Sports football series Soccer AM.