Greaves was born at 7A Rockville Street, Birkenhead. His father, Charles Edward Greaves, a post office official, and his mother, Amy Elisabeth Taylor, were Methodists, of mixed English, Irish and Welsh background. He studied at Liverpool University where he graduated in chemistry and botany, he worked as a research chemist at Powell Duffryn. In 1934 he joined Communist Party of Great Britain. In 1941 he joined the Connolly Club which became the Connolly Association, and became editor of its magazine, The Irish Democrat.