In his career, he played as a wicketkeeper for the national Trinidad and Tobago team as well as playing 62 Tests for the West Indies. He was first selected for the West Indies as a 20-year-old, under the captaincy of Frank Worrell; in his maiden series in 1963 he effected a record 24 dismissals. Though he never scored a Test century, Murray's elegant right-handed batting in the middle order could be effective. During his highest Test score of 91, against India in 1975, he shared a partnership worth 250 runs with Clive Lloyd (who scored 242 not out).
Murray served as a diplomat in the Foreign Service of Trinidad and Tobago 1978–1989, becoming a representative to the United Nations in New York, where he served as Vice-Chairman of the Fifth Committee and Chairman of the Committee for Programme & Coordination. He also worked in the financial services industry. Murray officiated as match referee for three One Day Internationals in 1992. He is currently serving as President of the Trinidad and Tobago Cricket Board.