Melville was educated at the university in his native city of Edinburgh, where he obtained his PhD. After a period (1933–40) at Cambridge University, Melville served as professor of chemistry at the University of Aberdeen (1940–48) and at Birmingham University (1948–56).

At this point in his career Melville moved mainly into science administration. He advised such bodies as the Ministry of Power and the Electricity Authority and from 1956 to 1965 served as secretary of the Department of Scientific and Industrial Research. He finally held the post of head of Queen Mary College, London, from 1967 until his retirement in 1976.

As a chemist, Melville is noted for his work on chain reactions involving free radicals, in which he followed up the ideas of Cyril Hinshelwood and Nikolay Semenov and showed experimentally that they were correct. Later he studied the kinetics of polymerization chain reactions. He wrote Experimental Methods in Gas Reactions (1938) with Adalbert Farkas.