Murray was born and raised in Bristol. Her parents, Rosemary Silvester and Martin Murray, both work at Bristol University. Her father is a professor and her mother is a research technician.

At the age of 16, Murray heard about an audition for young actors in Bristol, and decided to audition for the experience. The auditions were for the E4 teen drama series Skins. She impressed the producers of the series and was cast as Cassie Ainsworth, a gentle and creative but self-destructive teenager with an eating disorder. Murray went on to appear in the first two seasons, from 2007 to 2008, then all of the original cast left at the end of the show’s second season to make way for a new generation of characters. But until today, her portrayal of Cassie Ainsworth remains one of the greatest icons of the series.

Following Skins in 2008, Murray made her stage debut as Mia in the critically acclaimed That Face, a West End production at the Duke of York’s Theatre. She was highly praised for her acting in the play, alongside the cast, and it was considered a milestone in her career as she abandoned her well-known ‘Cassie way’. That same year, she had a small role in the black comedy In Bruges, but her scene was eventually cut from the final version of the film.