[1] His parents, David Burke and Anna Calder-Marshall, are also actors, as were his godparents, Alan Rickman and Bridget Turner.

[4] He attended dance school before being accepted at the Royal Academy of Dramatic Art (RADA) in London when he was 18.

[citation needed] His first television part after drama school was Syd in the Paul Abbott thriller series State of Play, starring John Simm, Bill Nighy and James McAvoy.

In 2007, he played Napoleon Bonaparte in an episode of the BBC's docudrama Heroes and Villains and had a small part as a book publisher in the satirical drama The Trial of Tony Blair.

In 2009, he played Geoff Goddard in Telstar: The Joe Meek Story, and had a small part in Stephen Frears' Chéri.

In 2013, he played Billy, the older brother of Ryan Gosling's character in Only God Forgives, directed by Nicolas Winding Refn.

[6] In 2020, he played American filmmaker Orson Welles in David Fincher's Netflix original film Mank, opposite Gary Oldman as Herman J. Mankiewicz.

He also starred in English director and photographer Mitch Jenkins's 2020 film The Show (written by Alan Moore) as private investigator Fletcher Dennis.

In November 2021, Burke joined Anya Taylor-Joy and Chris Hemsworth in the Mad Max: Fury Road spinoff film Furiosa: A Mad Max Saga, replacing Yahya Abdul-Mateen II, who had to drop out due to scheduling conflicts.