Everett was born and raised in Norfolk, England, the son of Sara (née MacLean) and Major Anthony Michael Everett, who worked in business and served in the military. Through his maternal grandparents, Lady Opre Vyvyan and Vice Admiral Sir Hector Charles Donald MacLean, he is a descendant of the baronets Vyvyan of Trelowarren and the German Schmiedern barons, as well as a grand-nephew of Donald Duart Maclean, the Soviet double agent, and a great-grandson of the Liberal politician Sir Donald Maclean, who was Leader of the parliamentary opposition in the years following the First World War. He has an older brother, Simon Anthony Cunningham Everett.

From the age of 7 he was educated at Farleigh School, Hampshire, and later was educated by Benedictine monks at Ampleforth College, Yorkshire, but he dropped out of school aged 15 and ran away to London to become an actor.

In order to support himself, he worked as a male prostitute, or "rent boy", for drugs and money as he later admitted to US magazine in 1997. After being dismissed from the Central School of Speech and Drama for insubordination, he travelled to Scotland and got a job at the Citizens' Theatre in Glasgow.

His break came with the 1982 West End production of Another Country, playing a gay schoolboy opposite Kenneth Branagh, followed by a film version in 1984 with Colin Firth. He began to develop a promising film career, until he co-starred with Bob Dylan in the huge flop Hearts of Fire (1987).

Around the same time, Everett recorded and released an album of pop songs, entitled Generation Of Loneliness. Despite being managed by the largely successful pop svengali Simon Napier-Bell (who also managed Marc Bolan, launched and managed Japan, and steered Wham! to international fame) and the title track reaching the Top 40 in the UK, the public didn't take to his change in direction.

The shift was shortlived, and he would only return to pop indirectly by providing backing vocals for his friend Madonna many years later, on her cover of "American Pie" and on the track "They Can't Take That Away from Me" on Robbie Williams' Swing When You're Winning in 2001.

In 1989 he moved to Paris, writing a novel Hello, Darling, Are You Working? and coming out as gay, a move which some at the time perceived as damaging to his career. Returning to the public eye in The Comfort of Strangers (1990), several films of variable success followed.

The Italian comics character Dylan Dog, created by Tiziano Sclavi, is graphically inspired by him. The English actor, in turn, later appeared in an adaptation of a novel based on Sclavi's novel, Dellamorte Dellamore. In 1995 he released a second novel, The Hairdressers of St. Tropez.

His career was revitalized by My Best Friend's Wedding (1997), playing Julia Roberts's gay friend. In 1999, he played Madonna's gay best friend in The Next Best Thing (he also sang backup on her cover of "American Pie", which is on the film's soundtrack). He has since appeared in a number of high-profile film roles, often playing heterosexual leads. He is also a Vanity Fair contributing editor.