Carson was born in Los Angeles, California, the son of Actress Karen Black and actor, Producer, Screenwriter and Director L. M. Kit Carson. He is the stepson of Stephen Eckelberry and stepbrother of Celine Eckelberry, Stephen's daughter. He made his first national appearance in October 1976, at the age of nine months, when his mother hosted Saturday Night Live. During her opening monologue, Black held Hunter in her arms the entire time. He is a graduate of Wesleyan University.

Carson's first acting role was in the 1984 film Paris, Texas, portraying the character of Hunter Henderson. The film Paris, Texas was adapted for the screen by his father, L. M. Kit Carson. Hunter Carson received strong reviews for his performance in the film. He then starred in the 1986 remake of Invaders from Mars, where he co-starred with his mother Karen Black, and appeared in the 1988 comedy drama Mr. North, which also featured his Paris, Texas co-star Harry Dean Stanton. He played Bud Bundy in an unaired early pilot of Married... with Children, and appeared as Peter McMichaels in the 2001 fashion drama film Perfume. He played a "crazy killer boyfriend" in the 2010 horror film She's Crushed. He co-directed with Alejandro Itkin the 2013 feature film Single in South Beach, a romantic drama starring Kevin Sorbo. He also directed the short With It (2004), about a failed hitman, which Andrew Dansby says echoes the offbeat yet deliberate and contemplative approach of Wenders in Paris, Texas.