Murphy is born to an Anglo-Irish family at Milford House, near the Mayo-Galway border. His childhood in Ireland is documented in the film The Other Irish Travellers made by his niece, Fiona Murphy.
He spends much of his early childhood in Ceylon, now Sri Lanka, where his father, William Lindsay Murphy, serves in the Colonial Service and is active as mayor of Colombo and Governor-General of the Bahamas, in succession to the Duke of Windsor. He first receives his education at Canterbury School and Wellington College, Berkshire. He wins a scholarship to Magdalen College, Oxford, at 17, where he studies English under C.S. Lewis. He is later educated at the Sorbonne and, between 1953 and 1954, he runs a school in Crete. In his Archaeology of Love (1955), Murphy reflects on his experiences in England and the Continent.
In 1954, Murphy settles at Cleggan, a village on the coast of Galway where fishing has been abandoned after a famous sailing disaster. Several years later, in 1959, he purchases and renovates the Ave Maria, a traditional Galway hooker type boat, from Inishbofin fisherman, Michael Schofield, which he uses to ferry visitors to the island. Taking the first-hand accounts of survivors of the sailing disaster, he weaves the material into a long tour de force poem which closes his first collection Sailing to an Island, published in the early 1960s by Faber & Faber. In 1969, he purchases Ardoileán (High Island), a small island in the vicinity of Inishbofin.
Murphy enjoys commissions for his poems from the BBC which prompts him to start on his long book-length sequence The Battle of Aughrim. Ostensibly about the 18th century triumph of Dutch-led Protestant forces over the Irish and French Catholic forces, the poem deals obliquely not only with the brewing strife in Ulster of the 1960s, but also with the issues of the Vietnam War. Its episodic structure is highly influential on poetic sequences subsequently published by Montague and Heaney.
Since 1971 Murphy has been a poet-in-residence at nine American universities. He lives in Sri Lanka, having previously divided his time between Dublin and Durban, South Africa, where his daughter and her family reside. He is the maternal grandfather of YouTuber Caspar Lee. In 2002, a memoir of his life and times, The Kick, is published by Granta, constructed from detailed diaries kept over the course of five decades.