Born in London, the son of a clergyman, Greene became a choirboy at St Paul's Cathedral under Jeremiah Clarke and Charles King. He studied the organ under Richard Brind, and after Brind died, Greene became organist at St Paul's.
With the death of William Croft in 1727, Greene became organist at the Chapel Royal, and in 1730 he became professor of music at Cambridge University. In 1735 he was appointed Master of the King's Musick. At his death, Greene was working on the compilation Cathedral Music, which his student and successor as Master of the King's Musick, William Boyce, was to complete. Many items from that collection are still used in Anglican services today.
Greene wrote a good deal of vocal music, both sacred and secular, including the oratorio The Song of Deborah and Barak (1732, settings of sonnets from Edmund Spenser's Amoretti (1739), and a collection of anthems (1743), of which the best-known is Lord, let me know mine end.