- He discovered fellow Liverpudlian Jodie Comer while filming scenes for one episode of Good Cop (2012) with her. Impressed by her talent, he immediately called his agent Jane Epstein and recommended the young actress to her. This led to a boost in Comer's acting career and to a friendship between both actors. When Comer won her BAFTA award in May 2019 for "Best Leading Actress" for her performance in Killing Eve (2018), she thanked Graham for introducing her to Epstein and making her career possible.
- His paternal grandfather was Jamaican, and one of his grandmothers was Swedish. Despite his Afro-Caribbean roots, he portrayed a white nationalist in This Is England (2006).
- His solid acting work also includes three fine screen portrayals of infamous gangsters, Baby Face Nelson in Public Enemies (2009), Al Capone in Boardwalk Empire (2010) and Anthony "Tony Pro" Provenzano in The Irishman (2019).
- He was awarded the OBE (Officer of the Order of the British Empire) in the 2023 King's New Years Honours List for his services to drama. He is an actor in London, England. He dedicates the award to his mother who has passed away.
- Has worked seven times with Vicky McClure in This Is England (2006), Filth and Wisdom (2008), This Is England '86 (2010), This Is England '88 (2011), Line of Duty (2012), This Is England '90 (2015) and The Secret Agent (2016).
- Has been in five Shane Meadows productions such as This Is England (2006), This Is England '86 (2010), This Is England '88 (2011), This Is England '90 (2015) and The Virtues (2019).
- Is the father of a daughter, named Grace, and a son, named Alfie.
- Brother of Aston Kelly.
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