The young British actress shot to prominence as one of the teenagers of provocative Channel 4 series Skins (playing Cassie) and she's since been ticking off the lifetime ambitions ever since.
Right now, she's promoting her latest film, God Help The Girl, the debut feature from Belle and Sebastian's Stuart Murdoch (released in cinemas next week), which finds her playing a young singer who dreams of being in a band and embarks on a journey of self-discovery over the course of a dream-like summer in Glasgow with two friends.
Murray describes making the film as "the happiest I've ever been in my life", explaining: "It was the summer and Glasgow is such a wonderful city. It was me, Emily Browning and Olly Alexander [a former Skins co-star] and we all got on so well. We were like this horrible little gang because we had so many in-jokes.
"And filming a musical was like the most incredible, fantastical dream of a job. Stuart Murdoch is a really special human being - really creative but also really kind and gentle. The film has a really low budget but everyone was working together. It was one of those very special experiences."
Murray also sings in the film, which is something she initially found daunting.
"I'm not a trained singer at all," she admits. "I've auditioned on occasion for proper musical theatre-type stuff but I can't read music and I wasn't particularly good at it. So, doing the songs in the film was challenging."
Again, she credits Murdoch - who also wrote the score - with providing the perfect environment in which to overcome any inhibitions.
"From day one, Stuart said that he wasn't interested in finding the best singer but the person who was right for the character. He was such a good teacher."
At 25, Murray is continually opening herself up to new experiences, no matter how far it takes her out of any comfort zone. But that fearless approach has been a hallmark of a career that, by her own admission, surprises her as much as it did her family.
She candidly admits, for instance, that her parents were surprised by her career choice - just as she was while juggling Skins with education.
"I remember my mum saying at one point, 'well, this is a really interesting point in your life, it'll be interesting to see what the next phase will be'. She couldn't imagine it carrying on as a career. And to be fair, I felt the same way. No-one in my family has ever been involved in this industry, even indirectly. Both myself and my parents thought it was very strange and didn't really understand it. I think we'd all always assumed that I'd do a normal job - that it was a fun holiday almost."
But while studying English at Cambridge University, Murray discovered how much acting meant to her.
"I came to realise that this was something I was passionate and serious about because I was doing it while it was quite inconvenient and difficult to fit in. And one of the things about the course is that they teach you to analyse texts in incredibly close detail. You can literally spend hours on one word if you want to - but that really, really helped with my understanding of scripts and the way I read them. I think I'm very, very good at reading things now and that is more of a skill that I have since completing Cambridge."
It's this eye for a good script that helped Murray land a role in hit TV show Game of Thrones, as Gilly. Describing her involvement with a HBO production as "a dream come true", she continues to be amazed by its popularity as well as its ability to surprise.
She remains tight-lipped about any potential future developments, including the fate of her character or that of her protector, Samwell Tarley (John Bradley), even though she does have an idea of who might be left standing come the show's end.
"My theory is that anyone that's weak actually does quite well. If you're a Ned Stark [Sean Bean] type of character you may not survive.
"Ned got killed in the first series because he was one of the types of characters that made everyone feel safe. If he was around, things might not get that bad - and then suddenly he was gone.
"Whereas Daenerys [Emilia Clarke] seemed an unlikely bet to survive when we first saw her because she was such a young girl and she seemed to have absolutely no reason to survive or be strong.
"So, I think the show really does favour the underdog. And Sam and Gilly are, in some ways, the most pathetic characters in it but their strength is in triumphing against the odds. So, I sort of have hopes for them because it's so unlikely they should survive, especially with a baby as well."
For now, Murray remains an integral part of the show as it enters its fifth season but she's also continuing to line up film projects, with another indie drama, Lily & Kat, completed along with Danish drama Suburbs, based upon the Bridgend suicides in South Wales. And she's savouring every minute.
"I remember before I started acting that it was always this impossible dream, so I feel really privileged that I get to do something I love as a job. And I think that's the best way you can hope to live your life - if you can spend the bulk of your time on something fulfilling, then that's a big thing and I feel really lucky to be in that position."
God Help The Girl opens on August 22. Special screenings will take place at Glasgow Film Theatre on August 16 and 17 and at the Cameo, Edinburgh, on August 16.