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.

After Iowa, Greg's first professional stage role was in The Death of Bessie Smith. One of his earliest television roles was a cameo appearance on The Dick Van Dyke Show in the episode "That's My Boy?", where Rob becomes convinced that they've taken home the wrong baby from the hospital. The revelation of Morris' character as the other child's father prompted a record setting bout of laughter from the studio audience. In the 1963–1964 season, he appeared on ABC's drama about college life, Channing, starring Jason Evers and Henry Jones.

After Mission: Impossible was cancelled, Morris appeared in movies and made guest TV appearances (1974 -The Six Million Dollar Man - Episode 1-05 "Little Orphan Airplane") before he was cast as Lt. David Nelson during the second season of the TV series Vega$. After the cancellation of that series in 1981, Morris continued to make guest TV appearances in the next decade, including a few episodes in the short-lived 1980s remake of the Mission: Impossible TV series, which starred his son Phil Morris, who was cast as Grant Collier, the son of Barney. He also appeared in two episodes of the TV series What's Happening!! as Lawrence Nelson (father of Dwayne) and in three episodes of The Jeffersons, in which he reprised his role of an electronics expert (although not as Barney Collier) in a comparison sequel of the Mission: Impossible series. Morris was also a frequent guest star on Password and Password Plus in the 1960s and 1970s.

Morris died on August 27, 1996 of brain tumor in Las Vegas, Nevada. He was 62 years old. Shortly before his death, he went to see the film version of Mission: Impossible that starred Tom Cruise. The reports were that he disliked the movie so much (an opinion that was shared by most of his former co-stars) that he left the theater early. According to the Associated Press, he said of the movie: "It's an abomination."