Clare was born in Dublin, in the Republic of Ireland and educated at Gonzaga College. In 1966, he graduated from University College, Dublin (UCD) where he was an auditor of the Literary and Historical Society. During his time in UCD, he won the 1964 Observer Mace debating competition, speaking in a team with Patrick Cosgrave. Following initial training in psychiatry at St Patrick's Hospital, Dublin, he moved to the Institute of Psychiatry at The Maudsley Hospital, in London where he studied under Professor Michael Shepherd. Clare held a doctorate in medicine and a master's degree in philosophy, and was a fellow of the Royal College of Psychiatrists.

Author of several popular books on psychiatry, Clare held the positions of Professor of Clinical Psychiatry at Trinity College, Dublin and Medical Director of St. Patrick's Hospital, Dublin. He was due to retire from his current post as Consultant General Adult Psychiatrist at St. Edmundsbury Hospital in Lucan, County Dublin when he died suddenly of a heart attack in Paris at the age of 64.

In the 1980s and 1990s, Clare was the best-known psychiatrist in Britain. His first media appearances were on the light-hearted BBC Radio 4 current affairs programme Stop the Week. He was also for many years the voice of the BBC popular science programme QED. Clare became famous for his probing interviews on radio and television with well-known figures such as Bob Monkhouse and Paddy Ashdown in several series of In the Psychiatrist's Chair, which ran from 1982.

As a young man Clare lost his Catholic faith and later explained why in a newspaper interview.

I can't really believe in a God that can suddenly and haphazardly intervene during one moment of history, causing air crashes, genocide and famine.

Clare married Jane Hogan in 1966 and they had seven children together.