Murray was born on 8 September 1959 in Bridge of Allan, Stirlingshire, the daughter of Eileen Shirley (née Edney) and Roy Erskine, an optician and former footballer who had played for Stirling Albion in the 1950s. She says that growing up, there were no indoor tennis courts in Scotland, so she played tennis in the summer and badminton in the winter. She won 64 titles in Scotland during her junior and senior career, and decided to have a go at the professional tour in around 1976. Murray gave up the idea of competing professionally as she was homesick and was robbed in Barcelona. However, she had played against players such as Debbie Jevans and Mariana Simionescu. Murray claims that her playing style did not have any big shots but she was quick around the court and read the game well. She decided to go to the University of Edinburgh to study French and German, before dropping German in favour of business studies. In 1981, she graduated from the University of Edinburgh, and represented Great Britain at the World Student Games. She began coaching and was the initial coach for both her sons before handing over the reins as their professional careers bloomed. In the early 1980s, Murray lived in the West End of Glasgow and was a member of Broomhill Lawn Tennis and Squash club , winning the Club Championships three times and playing for its teams (under her maiden name). She remains a visitor to the club.
Aside from her own sons, she has coached many players at regional and national level under the auspices of the British tennis governing body, the Lawn Tennis Association (LTA). In December 2011 she was elected to lead the British Fed Cup team as their captain. She says she took the job in part to raise the profile of female coaches and alleviate some of the sexism that she says remains in the sport. She resigned as Great Britain‘s Fed Cup Captain in March 2016.