Born in Cleveland, Ohio, to the jazz trumpeter Francis Williams, Morris served in the United States Army during the Korean War. Morris began his acting career in the 1960s making guest appearances on many TV shows such as The Twilight Zone, Branded and Ben Casey. In 1966, he was cast in his most recognizable role as the electronics expert Barney Collier in the TV series Mission: Impossible. Morris, Peter Lupus, and Bob Johnson were the only actors to remain with the series throughout its entire run.