Trevelyan was the second son of Sir George Trevelyan, 2nd Baronet and his wife Caroline née Philips, who was the daughter of Robert Needham Philips MP, a Liberal Member of Parliament and textile merchant from Lancashire. Trevelyan was the brother of Sir Charles Trevelyan, 3rd Baronet, and the historian G. M. Trevelyan.

He was born in Weybridge, and educated at Harrow. From 1891 to 1895 he studied at Trinity College, Cambridge, being one of the Cambridge Apostles. He studied classics and then law; his father wanted him to follow a career as a barrister, but his ambition was to be a poet.

Described as a "rumpled, eccentric poet", and sometimes considered a rather ineffectual person, he was close to the Bloomsbury Group, who called him 'Bob Trevy'. He had a wide further range of social connections: George Santayana from 1905; Isaac Rosenberg; Bernard Berenson; G. E. Moore; E. M. Forster with whom he and Goldsworthy Lowes Dickinson travelled to India in 1912. His pacifist principles extended to sheltering John Rodker during World War I.

He married the Dutch musician Elizabeth van der Hoeven; the artist Julian Trevelyan was their son.

Trevelyan wrote a number of verse plays; The Bride of Dionysus (1912) was made into an opera by Sir Donald Tovey.