Cox, the third of five children, was born in Cloudcroft, New Mexico, the son of Lounette (née Rucker) and Bob P. Cox, a carpenter who also worked at a dairy. He grew up in Portales, New Mexico. Cox met his wife Mary when she was in the fifth grade and he was in the seventh grade, and they married in 1960 and have two children. Cox graduated from Eastern New Mexico University in 1963 with a double major in theater and speech correction. Mary died in 2006, 50 years to the day of their first date. Cox often talks about her during his music performances.

In 1974–1975, Cox starred in the short-lived family-oriented series entitled Apple’s Way, created by Earl Hamner, creator of The Waltons. He also appeared as Mr. Webb in a television production of Our Town. In 1977, he appeared in the episode “Devil Pack” from the series Quinn Martin’s Tales of the Unexpected. In 1984, 12 years after Deliverance, Cox again played a member of a small group of men who are lost, this time in the Nevada desert, and being chased by bloodthirsty locals in the low-budget film Courage.

Cox’s first role in a big-budget film came in 1984 as Lt. Andrew Bogomil in Beverly Hills Cop, and he returned to the role in Beverly Hills Cop II in 1987.

In 1990, Cox co-starred as Los Angeles Police Chief Roger Kendrick in the short-lived Cop Rock, presenting a striking physical resemblance to the real-world incumbent Chief Daryl Gates. He also appeared as the antagonistic Mars Administrator Vilos Cohaagen in Total Recall the same year.

In 1997, Cox portrayed the fictional President of the United States Jack Neil in the movie Murder at 1600. Cox also portrayed John Ramsey in the 2000 TV film Perfect Murder, Perfect Town and Senator/Vice President Robert Kinsey in Stargate SG-1.

As an actor, Cox made his debut in the acclaimed 1972 film, Deliverance. In one scene, he plays the instrumental “Dueling Banjos” on his guitar with a banjo-playing mountain boy, played by child actor Billy Redden. He was hired for the role because he could play the guitar. Cox published his autobiography in 2012, recounting his experiences making the film.

In 2018, Cox played Gideon Claybourne on season 6 of Nashville.