Cox was born in Bebington, Cheshire, England in 1954. He attended Worcester College, Oxford, and later transferred to the University of Bristol where he majored in film studies. Cox secured a Fulbright Scholarship, allowing him to study at the University of California, Los Angeles, where he graduated from the School of Theater, Film and Television with an MFA.

Cox began reading law as an undergraduate at Oxford University, but left to study Radio, Film and TV at Bristol University, graduating in 1977. Seeing difficulties in the British film scene at the time, Cox first went to Los Angeles to attend film school at UCLA in 1977. Here he produced his first film, Edge City/Sleep is for Sissies, a 40-minute surreal short about an Artist struggling against society. After graduation, he formed Edge City Productions with two friends with the intention of producing low-budget feature films. Since that day Cox decided to dedicate his life for films.

Cox has cited Luis Buñuel and Akira Kurosawa as influences, as well as the great Western film Directors Sergio Leone, Sam Peckinpah and John Ford. Cox also wrote a book on the history of the genre called 10,000 Ways to Die. While he once directed films for Universal Pictures, such as Repo Man and Walker, since the late 1980s, he has found himself on a self-described blacklist, and turned to producing independent films. Cox is an atheist and is decidedly left wing in his political views. Many of his films have an explicit anti-capitalist theme or message. He was originally set to direct Fear and Loathing in Las Vegas but was replaced by Terry Gilliam due to creative differences with Hunter S. Thompson. By August 2009, Cox had announced completion of Repo Chick, which premiered at the Venice Film Festival the following month, but he remained ambivalent as to whether the film would ever be distributed to cinemas. His previous film, Searchers 2.0, was not released theatrically, and only appears on DVD in Japan and North America after a televised screening in the UK on the BBC.

In May 1988 Cox began presenting the long-running and influential BBC series Moviedrome. The weekly strand was a showcase for cult films. Though most of the films shown were chosen by series creator and Producer Nick Jones, each film was introduced by Cox. By the time he left the show in September 1994, Cox had introduced 141 films. Various film Directors have cited Moviedrome as an influence, including Ben Wheatley and Edgar Wright. The series was later presented by film Director and critic Mark Cousins.

Shortly after this, Cox was invited to adapt a Jorge Luis Borges story of his choice for the BBC. He chose Death and the Compass. Despite being a British production and an English language film, he convinced his producers to let him shoot in Mexico City. This film, like his previous Mexican production, made extensive use of long-takes. The completed 55-minute film aired on the BBC in 1992.

Cox had hoped to expand this into a feature-length film, but the BBC was uninterested. Japanese Investors gave him $100,000 to expand the film in 1993, but the production ran over-budget, allowing no funds for post-production. To secure funds, Cox directed a "work for hire" project called The Winner. The film was edited extensively without Cox's knowledge, and he had his name removed from the credits as a result, but the money was enough for Cox to fund the completion of Death and the Compass. The finished, 82-minute feature received a limited cinema release in the US, where the TV version had not aired, in 1996.

In 1996, Producer Stephen Nemeth employed Alex Cox to write and direct an adaptation of Hunter S. Thompson's Fear and Loathing in Las Vegas. After creative disagreements with the Producer and Thompson, he was sacked from the project, and his script rewritten when Terry Gilliam took over the film. (Cox later sued successfully for a writing credit, as it was ruled that there were enough similarities between the drafts to suggest that Gilliam's was derivative of Cox's. Gilliam countered that both screenplays were based directly on the source book, and any similarities between the two screenplays were a direct consequence of this.)

In 1997, Alex Cox made a deal with Dutch Producer Wim Kayzer to produce another dual TV/feature production. Three Businessmen. Initially, Cox had hoped to shoot in Mexico, but later decided to set his story in Liverpool, Rotterdam, Tokyo and Almería. The story follows businessmen in Liverpool who leave their hotel in search of food and slowly drift further from their starting point, all the while believing they are still in Liverpool. The film was completed for a small budget of $250,000. Following this, Cox moved back to Liverpool and became interested in creating films there.

Cox is a fan of the Japanese Godzilla films and appeared in a 1998 BBC documentary highlighting the series. He also narrated the documentary Bringing Godzilla Down to Size and wrote the Godzilla in Time comics for Dark Horse. He tried to direct an American Godzilla film at one point, but unsuccessfully submitted his outline to TriStar Pictures.

In 2006, Alex Cox tried to get funding for a series of eight very low budget features set in Liverpool and produced by locals. The project was not completed, but the Director grew interested in pursuing the idea of a film made for less than £100,000. He had originally hoped to shoot Repo Man on a comparable budget, and hoped that the lower overhead would mean greater creative freedom.

In a March 2007 blog post, Cox referred to Vice President Dick Cheney as "secret Architect of the 9-11 atrocities." In the same article, Cox called the September 11 attacks "Plan Pearl Harbor," referring to the false flag conspiracy theory surrounding the attack on Pearl Harbor.

Alex Cox had attempted to get a Repo Man sequel, titled Waldo's Hawaiian Holiday, produced in the mid-'90s, but the project fell apart, with the script adapted into a graphic novel of the same name. For his next micro-feature, he wrote a fresh attempt at a Repo follow-up, although it contained no recurring characters, so as to preserve Universal's rights to the original. Repo Chick was filmed entirely against a green screen, with backgrounds of digital composites, live action shots, and miniatures matted in afterwards, to produce an artificial look. It premiered at the Venice Film Festival on 9 September 2009.

Searchers 2.0, named after but based on The Searchers, became Cox's first film for which he has sole writing credit since Repo Man, and marked his return to the comedy genre. A road movie and a revenge story, it tells of two actors, loosely based on and played by Del Zamora and Ed Pansullo, who travel from Los Angeles to a desert film screening in Monument Valley in the hopes of avenging abuse inflicted on them by a cruel Screenwriter, Fritz Frobisher (Sy Richardson). It was scored by longtime collaborator Dan Wool aka Pray for Rain (Sid & Nancy, Straight to Hell, Death & the Compass, The Winner, Three Businessmen, Repo Chick among others). Although the film was unable to achieve a cinema release in America or Europe, Cox claimed the experience of making a film with a smaller crew and less restrictions was energising. It is available on DVD in Japan, and was released in October 2010 in North America.

As of 2011, Cox resided in Colestin, Oregon with his wife, Writer Tod Davies.

As of July 2012, Cox was teaching film production and screenwriting at the University of Colorado at Boulder.

In 2013 Cox directed Bill, the Galactic Hero, developed from a science fiction book by Harry Harrison. It was funded by a successful Kickstarter funding campaign, raising $114,957 of the original $100,000 goal. The film was to be made, created and acted by his film students in monochrome with supervision from professional film makers who would be giving their time on the film for free.

Cox’s 2013 book The President and the Provocateur examines events in the lives of John F. Kennedy and Lee Harvey Oswald leading up to Kennedy’s assassination, with reference to the various conspiracy theories.