Fletcher trained at the Anna Scher Theatre drama club. Fletcher's first film part was as Baby Face in Bugsy Malone (1976). He made his stage début the following year in a production of A Midsummer Night's Dream. As a child actor, Fletcher was a regular feature in British productions in the early 1980s such as The Long Good Friday, The Elephant Man, and The Bounty. As an adult, he has appeared on television as rebellious teenager Spike Thomson in Press Gang and also Murder Most Horrid (1991) with Dawn French. His American accent had some fans convinced that he actually is an American. He has starred in the films Caravaggio (1986), The Rachel Papers (1989), Lock, Stock and Two Smoking Barrels (1998), Layer Cake (2004), AffirmFilm's Solomon as Rehoboam, and Universal's Doom as a communications officer nicknamed "Pinky".

Fletcher has been the voice for McDonald's television adverts and (feigning a US accent) is the narrator of The Game audio book written by Neil Strauss. He also narrated the Five series Airforce Afghanistan, as well as the Chop Shop: London Garage series on the Discovery Channel. In 1993 he was the voice of Prince Cinders. In 1998, Fletcher featured on the song "Here Comes the Flood" from the album Fin de Siecle by The Divine Comedy. In 2014 he narrated the BBC1 show Del Boys and Dealers.

Fletcher recalls presenting GamesMaster was 'very unforgiving' and 'quite intimidating'. He claimed the most difficult part was 'just getting up, taking a deep breath, and going out there and doing it'. His style of presenting was heavily criticised by viewers who said his 'in your face' method was too 'OTT' and did not suit the show. Some viewers suspected his genuine Cockney accent was put on for effect. The series finished in 1994 and Dominik Diamond returned for the last four series until the show ended in February 1998. The show's creator Jane Hewland later said in a retrospective interview, that she sees the third series as a "lost" one, due to admitting to a series of poor decisions and mistakes, including hiring Fletcher as main presenter. His name was however included on a rolling list of acknowledgements, which replaced the usual end credits when the final episode was broadcast, but has not taken on any other presenting role before or since.

Fletcher was born in Enfield, England. He dated Press Gang co-star Julia Sawalha and also had a relationship with Actress Liza Walker. In 1997 he married Lithuanian film and theatre Director Dalia Ibelhauptaitė in Westminster. His best man was fellow actor Alan Rickman. Dexter's brothers are also actors; Graham Fletcher-Cook and Steve Fletcher.

On television, he has appeared in the major HBO drama, Band of Brothers and in a supporting role in the BBC One historical drama The Virgin Queen (US PBS 2005, UK 2006). He also appeared in Kylie Minogue's 1997 released music video for "Some Kind of Bliss". Fletcher has starred on BBC One in a series based on the Imogen Edwards-Jones's book, Hotel Babylon which ran for four series before being cancelled in 2009. He also appeared in one episode, "The Booby and the Beast", of BBC's Robin Hood series 2 as the German Count Friedrich and in the 2008 radio series The Way We Live Right Now.

He also appeared in the Boon episode "Walking Off Air", playing Eddie Cotton. In 2009, he appeared in Misfits as Nathan Young's dad, reprising the role in 2010 for the second series. Fletcher stars in 2009 in the vampire film Dead Cert, which is directed by Steven Lawson.

2011 saw him acting in his first role as a police officer, as Detective Edwards in the British film noir Jack Falls, alongside his Lock, Stock and Two Smoking Barrels co-stars Jason Flemyng and Alan Ford. In 2012, Fletcher had a small role in horror-comedy Cockneys vs Zombies, where he played the character 'Dad Maguire'. In 2015 Fletcher played the role of 'Scooter' in episode 5 of the 8-part BBC TV drama The Interceptor. He also took the lead role as history Teacher Mr Sheers in supernatural British teen film Coven. In 2016 Fletcher played the role of 'Paul' in the cult British crime comedy, Smoking Guns.

Fletcher's debut as a Director, Wild Bill, was released on March 20, 2012. His second film as Director is a musical film by Stephen Greenhorn, Sunshine on Leith based around the popular Proclaimers songs which was released on 4 October 2013. In 2016, he directed the feature film Eddie the Eagle. On December 6, 2017, Fletcher was announced as Bryan Singer's replacement in the upcoming Queen biopic, Bohemian Rhapsody, to be released on November 2, 2018.