Chaplin was born in Windsor, Berkshire, England, the son of Cynthia, a teacher, and Peter Greenwood CBE, a civil engineer. He has one sister, Rachel, and one brother, Justin. Chaplin became interested in acting as a teenager, after acting in a theatrical production in his school years at the Princess Margaret Royal Free School. At the age of seventeen, he enrolled at the Guildhall School of Music and Drama in London. He pursued his early acting career between odd jobs as an office worker, and for a while was employed as a statistician with the London Transport Authority.

Chaplin made his professional acting debut in the 1990 television film Bye Bye Baby. He went on to appear in a number of other television films and miniseries, including The Bill (1990), The Final Cut (1995), and The Lost World (1999).

Chaplin's breakthrough film role came in 1996, when he starred opposite Uma Thurman and Janeane Garofalo in the romantic comedy The Truth About Cats & Dogs. The film was a critical and commercial success, and it helped to establish Chaplin as a rising star.

Chaplin has since gone on to star in a number of other successful films, including Washington Square, The Thin Red Line, Birthday Girl, Murder by Numbers, Stage Beauty, The New World, The Water Horse: Legend of the Deep, Dorian Gray, Cinderella, Snowden, The Legend of Tarzan, and The Dig.

Chaplin has also had a successful career on television. He starred in the BBC sitcom Game On (1995–1998), and he has also appeared in the television series Mad Dogs (2011–2013) and The Nevers (2021–present).

In addition to his acting career, Chaplin is also a musician. He plays the guitar and the piano, and he has written songs for a number of films.

Chaplin is married to the actress Amanda Abbington. They have two children together.