[1] Murray was born in Hampstead, London, the daughter of Labour Party politician, Tony Greenwood, who was a member of Harold Wilson's cabinet.

Her mother Jill Greenwood, was a writer who created and illustrated the pamphlet Make Do and Mend, published during World War II, and was co-founder of the Campaign for Nuclear Disarmament.

She worked for Penguin Books as a copy editor before attending University College London, where she earned a degree in Linguistics and Anthropology in 1969.

[3] During those years, Murray collaborated with autistic associates Wenn Lawson and Mike Lesser, to develop the theory of "monotropism", an attempt to explain autism in terms of a tendency to focus on a single subject at a time.

[3] After the 2019 United Kingdom general election returned a Conservative majority, she moved to Dalgety Bay in Fife, Scotland.