Murray was born in Musselburgh, Scotland. She won the bronze medal at the 1986 Commonwealth Games in Edinburgh for the 3000 m. She then went on to win the bronze medal at the 1986 European Championships. In 1987, she was the European indoor champion at 3000 m. At the 1988 Olympic Games in Seoul, she won a bronze medal for the 3000 m. In 1989, she won the World Cup 3000 m in. In January 1990, at the Commonwealth Games in Auckland, she won silver in the 3000 m behind Canada's Angela Chalmers, with fellow Scot Liz McColgan third. Later that year, she won a gold medal at the European Championships in Split in the 3000 m. In 1993, Murray won a gold medal at the World Indoor Championships, again over 3000 m. Then in 1994, she won silver at the European Championships, and then went on to win a gold medal in the Commonwealth Games, this time for the 10,000 m.
Murray was five-time Scottish outdoor champion. She was a UK champion at 3000 metres 1985 and 1987, and at 5000 metres in 1983. She was five-time AAA champion outdoors, and AAA indoor champion in 1984.