He attended Sturry Secondary Modern School.
Goffe appeared as an Oompa-Loompa in the 1971 version of Willy Wonka & the Chocolate Factory and as a Jawa in Star Wars Episode IV: A New Hope, among a few other aliens. He also appeared in the films Willow and Flash Gordon. He played Le Muff in the film History of the World Participant I and also played Goober, a purple gremlin butler in In 2002, Goffe played the lead role in A Kitten for Hitler, a film created by Ken Russell with the intention of making something as offensive as possible.
He played a Jewish child who is made into a lamp by Hitler.
He is also notable for being the face of Ginsters.