Morris was a church-choir singer from his youth, trained at the Juilliard School of Music, and soloed early in his career with the Harry Belafonte singers. He performed in numerous musicals.

Morris has appeared in numerous television shows and movies since the early 1970s, but is best known as one of the original cast members of NBC's Saturday Night Live. At least twice on SNL he sang classical music: once a Mozart aria when guest-host Walter Matthau designated him as a "musical guest...in place of the usual crap", and once a Schubert lied while the titles on the screen purported to express his colleagues' displeasure at having to accommodate a misguided request by him.

Morris' best known character on SNL was the Dominican baseball player Chico Esquela. Chico spoke only limited and halting English, so the joke centered on him responding to almost any question with his catch phrase: "Baseball... been berra berra good... to me." In the mid '70s Morris appeared as a high school teacher in the memorable cult classic film "Cooley High". During his later seasons on SNL, Morris allegedly freebased cocaine and suffered from severe delusions as a result. Morris was degraded from years of small roles and racist sketches (at one point the writers were going to have him do a fake ad for "Tar Baby" toothpaste, which would make blacks' teeth stop glowing in the dark -- only when black crew members walked off the set in protest did SNL producer Lorne Michaels drop the idea). During rehearsals for the Kirk Douglas hosted episode, he reportedly ran screaming onto the set, saying that someone had put an "invisible robot" on his shoulder who watched him everywhere he went. He pleaded with them to get the robot off of him.

In 1994 he was shot in an apparent robbery attempt but went on to make a full recovery.

Among later television performances, he had regular roles on Hunter, Martin, and The Jamie Foxx Show.

As of 2004, he continues to perform regularly in films.