Dillon attended Bournemouth School for Girls and left in 1997. She began a degree in English Literature and Drama at Royal Holloway, University of London but abandoned it when she won a place at the Royal Academy of Dramatic Art (RADA).

In 2003 she was nominated for the Evening Standard Award for Outstanding Newcomer of the Year for the performance she gave in a production of Ibsen's The Master Builder. She won that year's Ian Charleson Award, in part, for that performance.

In 2007, she returned to the Crucible to play Celia in Shakespeare's As You Like It during February (in a production that also played at the RSC complete works festival) and Varya in Chekhov's The Cherry Orchard during March. She then starred in the National Theatre's revival of Noël Coward's Present Laughter and The Hour We Knew Nothing of Each Other. She also appeared in Anna Mackmin's 2008 West End revival of Under the Blue Sky by David Eldridge.

In 2008 Dillon duetted with British singer-songwriter Tim Arnold on his song "She's Made A Gentleman Of Me".

In 2009, Dillon starred in When the Rain Stops Falling at the Almeida, and in 2010 in Design for Living and its successor production A Flea In Her Ear at the Old Vic. Eldridge created the role of Lucy in The Knot of the Heart, presented at the Almeida in March 2011, specifically for Dillon. In early 2012, Dillon played the role of Katharina in the Royal Shakespeare Company (RSC) Stratford and touring production of The Taming of the Shrew.

In 2011 Dillon read a Jarvis and Ayres production of Hilda Richards' 1939 story "Jemima Gets Them Guessing" for BBC Radio 4's Afternoon Reading show, on 9 March.

In 2014, she starred as Moll Cutpurse in The Roaring Girl with the RSC. In December 2015 she took the title role in Hapgood for Director Howard Davies at the Hampstead Theatre.