Kelly grew up in Barnet, North London, in an Irish family and was brought up a Catholic. His father was a bus conductor, and Kelly, one of five children, left school at 16.

Brought up on a council estate he said "There is a theatre version of a council estate, and it doesn't look like any council estate I've ever been on. Council estates are really diverse, especially in London. You get all sorts of people. Working-class people have the same vocabulary as anyone else. Some read the Guardian, some read the Sun, some read the Telegraph." About growing up on a council estate Kelly has said;

"A couple of years ago, when I was starting out as a playwright, I had a meeting with an artistic director about a play I had just written, Fifty-Three Million Miles. It was set partly on a council estate, partly in a Nasa interview room, and partly in a living pod on the surface of Mars. "You do good council estate," the artistic director told me. "Stick to council estate."I was a little taken aback, but I took a deep breath and told him that although, yes, I'd grown up in a council house, I didn't feel that it was necessarily all there was to me, and that there were many things I was interested in talking about.."

While working in supermarkets, he discovered theatre when he joined a local youth group, the Barnet Drama Centre. " He then took a degree in Drama and Theatre Arts and received a first from Goldsmiths College, London. He wrote his first play Debris when he was 30.

His plays include Debris (Theatre 503, 2003/ Battersea Arts Centre, 2004), Osama the Hero (Hampstead Theatre, 2005), After the End produced by Paines Plough at the Traverse/ Bush Theatre, 2005/ UK and international tour 2006), Love and Money (Royal Exchange, Manchester and Young Vic, 2006) and Taking Care of Baby (Hampstead Theatre, 2007). In 2007 he also wrote DeoxyriboNucleic Acid, written for the National Theatre Connections Festival in 2007. His most recent play Orphans was staged in 2009 at the Birmingham Repertory Theatre and transferred to Traverse Theatre in Edinburgh in August 2009 as part of the Edinburgh Fringe Festival.

His work has been produced in Germany, Austria, Switzerland, Slovakia, Holland, Ireland, The Czech Republic, Poland, Italy, Australia, Japan, the United States, Belgium, Romania and Canada. Other work includes translations of Péter Kárpáti’s The Fourth Gate (National Theatre Studio) and The Colony, a radio play which won Best European Radio Drama at the Prix Europa, 2004.

He also co-wrote the BBC Three sitcom, Pulling with writer/actress Sharon Horgan.

Plays

• Debris 2003
• Osama the Hero 2005
• After the End 2005
• Our Teacher's a Troll
• Rose Bernd
• Love and Money 2006
• Brendan's Visit
• Taking Care of Baby 2007
• D.N.A part of National Theatre Connections 2007
• Orphans 2009
• The Gods Weep 2010
• Matilda, A Musical 2010

Translations

• The Fourth Gate original play by Peter Karpati

Radio

• The Colony, 2004

Television

• Pulling for BBC Three co written with Sharon Horgan

Books

• Love and Money (Oberon Modern Plays, April 2007) ISBN 1-84002-695-2
• After the End (Oberon Modern Plays, April 2006) ISBN 1-84002-580-8
• Debris (Oberon Books, September 2004) ISBN 1-84002-433-X

Kelly said that writing for TV and theatre is very unusual as Pulling is a comedy and not theatrical unlike his plays which are serious and often non naturalistic. Kelly said "telling people from the world of TV that I also inhabit the world of theatre is something I've begun to avoid."Despite very good reviews and good ratings Pulling was cancelled in 2007. The decision by the BBC was much criticised and Kelly and Horgan claimed to have cried and threw themselves at their feet over the decision. Most striking about Pulling is its lack of a moral centre. In an interview with The Guardian Horgan said "I guess there isn't a moral centre because Dennis and I don't have one." Kelly then said "That's scary. Fuck. We need to get a moral centre. Shit. It's really true. But we do try to make sure we don't get nasty for the sake of it. We make sure there's a bit of heart. "

Consistent traits in Kelly's work is strong woman, violence, fantasy and flexible identities. For DNA Kelly said in an interview

This concept of a flexible identity was something I wanted to explore in a play. DNA was originally written for the National Theatre's Connections project, which pairs young actors with new writing. It was to be performed by more than 40 different youth groups across the country, and when I wrote it, I stipulated that all the characters' genders and names could be changed according to the groups' needs. John could become Jane, or Leah could become Lee. I reasoned that there isn't the huge gap between men and women that we like to think there is. We are different, yes, but our similarities far outweigh our differences. One quite angry youth leader took me to task over this, insisting that girls and boys were practically different species, and this could never work. But the interesting thing was that, with all the different cast configurations I went on to see, I forgot the original sex of the character I'd written within 10 minutes.