Dyson was born in Leeds, Yorkshire, England, son of Elaine nee Saville and Malcolm Dyson. He went to Leeds Grammar School which he found to be a "very unimaginative place". He studied Philosophy at the University of Leeds and later completed an MA in screenwriting at the Northern School of Film and Television. He lived in Highbury, London, but now lives in Ilkley, West Yorkshire.

He is best known as a member of the sketch comedy team The League of Gentlemen, along with fellow performers Mark Gatiss, Steve Pemberton, and Reece Shearsmith, all of whom he met at Bretton Hall drama school in his late teens. The League of Gentlemen initially began as a stage act in 1995, transferred to BBC Radio 4 as On the Town with the League of Gentlemen in 1997 and then arrived on television on BBC Two in 1999. The latter has seen Dyson and his colleagues awarded a British Academy Television Award, a Royal Television Society Award and the prestigious Golden Rose of Montreux.

He has written several books including Bright Darkness: Lost Art of the Supernatural Horror Film, a non-fiction guide to horror films, and two collections of short stories entitled Never Trust a Rabbit and The Cranes That Build The Cranes. His novel What Happens Now was published on 6 April 2006 to favourable reviews in The Guardian and The Independent.

He has co-created (with Simon Ashdown) the television series Funland, which aired on BBC Three, and wrote the Billy Goats Gruff episode of the BBC's 2008 series Fairy Tales.

Due to his self-confessed lack of acting skills and a rather camera-shy nature, he does not appear in The League of Gentlemen television series or any of its offshoots, apart from very brief cameos. He works as the assistant producer instead. In the film adaptation he is played by Michael Sheen although Dyson appears in the background of a few scenes.

Dyson has an interest in the supernatural fiction of English writer Robert Aickman and has adapted Aickman's work in a number of media.

Alongside his writing work, Dyson also plays keyboards in a pop band called Rudolf Rocker. They are based in Leeds and signed to Mook Records. Their best known album, The Exotic Sounds of Rudolf Rocker, features the single "Samba Suit". They can be found on the internet at http://www.myspace.com/rudolfrocker. Dyson had previously been a member of Leeds band Flowers For Agatha in the 1980s.

He is also a patron of the charity No Panic.

With Andy Nyman, he co-wrote and co-directed the supernatural-themed stage production Ghost Stories. The play had its first run at the Lyric Hammersmith, in London, between 24 April and 17 February 2010.