Graves first came to prominence in costume-drama adaptations of E. M. Forster's novels A Room with a View (1985) and Maurice (1987), before going on to appear in films including A Handful of Dust (1988), the Oscar nominated The Madness of King George (1994), Different for Girls (1996), and Intimate Relations (1996).

In 1987 in his hometown of Weston-super-Mare, Graves met Yvonne, a stained glass Artist (later a trained gardener), in a café. They lived together in Stoke Newington, and he helped her raise her two daughters, who were 10 and 14 years old when the relationship began. Graves and Yvonne were together for 13 years.

Graves's notable London theatre credits includes his performance as Presley Stray in the original production of Philip Ridley's The Pitchfork Disney (1991) at the Bush Theatre, west London, which won him Best Actor at the 1991 Charrington London Fringe Awards; Martin Sherman's A Madhouse in Goa (1989) opposite Vanessa Redgrave; and as Eddie in the Peter Hall Company's production of David Rabe's Hurlyburly (1997-8) at the Old Vic and Queen's Theatre, for which he was nominated for the 1998 Laurence Olivier Award for Best Actor.

Graves's role in Intimate Relations won him the Best Actor award at the 1996 Montreal World Film Festival. He was also acclaimed for his portrayal of Young Jolyon Forsyte in the television miniseries The Forsyte Saga (2002).

In addition to his screen work, Graves has also won acclaim for his stage acting, including roles on the American stage in Broadway-theatre productions in New York City, New York, of the plays Closer (2000) and The Elephant Man (2002).