Greenwood was born in Noranda, Quebec to Mary Sylvia Ledingham, a nurse who worked in an extended care unit, and Hugh John Greenwood, a Vancouver-born geophysicist and teacher who taught at Princeton University. He has two sisters, Kelly Louis and Barbara Lynn. Greenwood studied philosophy and economics at the University of British Columbia and also attended the Magee School.

Greenwood is known in the U.S. for his appearances in I, Robot, The Core, Thirteen Days (in which he played President John F. Kennedy), Capote (in which he played Jack Dunphy, Truman Capote's lover), and more recently Eight Below (in which he played Professor Davis McClaren). He has had prominent roles in the award-winning films Exotica and The Sweet Hereafter, both directed by Canadian director Atom Egoyan. He also appeared in the 1980's teen cult film The Malibu Bikini Shop, and starred in Mee-Shee: The Water Giant.

On television, Greenwood has had roles on St. Elsewhere (Dr. Seth Griffin, 1986-1988), Knots Landing (Pierce Lawton, 1991-1992) and the UPN series Nowhere Man (Thomas Veil, 1995-1996).

On June 10, 2007, a new show on HBO from the creator of Deadwood called John from Cincinnati premiered starring Greenwood.

Greenwood met his wife, Susan, in Canada when they were both 15. They have been married since 1985.