Kelly studied at Cambridge University and UCLA Film School.

He has appeared in The Pitmen Painters at the National Theatre and A Busy Day in London's West End and in New York in his own one-man plays and also in the US premiere of Ron Hutchinson's Beau Brummell in the title role. He was nominated for Best Actor for his work in Tom Stoppard's Arcadia (Manchester Drama Awards)

Kelly has published biographies of Antonin Careme, Beau Brummell and Casanova and written for most of the British broadsheets and the New York Times. He is a contribuiting editor of Food Arts Magazine.

The BBC Television drama This Charming Man was based on his biography. Other TV work includes Dennis Potter's Cold Lazarus, Drop the Dead Donkey, Silent Witness, Just William, Catherine Cookson's The Moth, Sensitive Skin and Time Trumpet.

Kelly's film work includes Merchant-Ivory's Howards End, Richard Attenborough's In Love and War and the Russian films Admiral and Alexei Balabanov's Voyna shot in Chechnya and the Caucasus for which Ian was nominated for Best Actor at the Montreal International Film Festival.

He has been cast as Hermione Granger's father in Harry Potter and the Deathly Hallows.

His biography of Beau Brummell was shortlisted for the Marsh Biography Award.His biography of Giacomo Casanova was The Sunday Times Biography of the Year 2008-9.