'Sorry I'm late," Kevin Eldon apologises as he sits down in his agent's office in London. He isn't, really, only a minute – though in another sense you could say Eldon is running about 20 years late. For the past two decades, he has featured in virtually every landmark British comedy series – Nighty Night (Julia Davis's husband), Big Train, I'm Alan Partridge (a racist salesman), Blue Jam – without having a starring role of his own. This weekend, he gets his own television series – a sketch-variety show called It's Kevin, complete with staircase and jaunty theme tune.

Eldon says the idea of having his own show hadn't entered his mind until two years ago, when he performed for the first time on the Edinburgh fringe. Until then, "I was always easy turning up at other people's parties, doing a little dance, then going home." The Edinburgh show meant that "a couple of bods from the BBC whispered into my agent's ear", and then commissioned six episodes of precision-tooled lunacy drawing on his live set (a rapping insurance salesman, a Hitler with Beatles producer George Martin's patrician voice), as well as new material.

The mix ranges from the accessible to the warped. A skit involving a man suffering from Soundtrackitis (appropriate music always accompanying his speech) could have come direct from The Two Ronnies. But then it gets more subversive. A pair of soiled pants or a cheesegrater will brush past Eldon's face; a cartoon Yorkshireman pitches up to talk about one of the Supremes waiting for a man to fix her boiler. There is terrific support from old pals, such as Davis and Amelia Bulmore. Eldon's horribly hummable Europop duet with Bill Bailey, Mobile Phone, looks destined to be a YouTube hit.

Reading this on mobile? Click here to view

In a youth-based business, Eldon's belated breakthrough is refreshing. "Fifty-three-year-old men don't get their first series on TV very often. But it's probably 50 times harder for women as far as age is concerned. You can't really moan about ageism as a bloke, when women are getting it in the neck all the time." Is he prepared for fame? "At the moment I have a level of recognition which is pleasant. I have a couple of mates who are really famous and I don't envy them, getting stopped every 200 yards to have their photo taken." How far can he walk without being recognised? "The length of Britain."

He recently worked with Martin Scorsese, cropping up as a gendarme in Hugo. "Sacha Baron Cohen was doing a scene where he phoned the police department and had the idea we should meet the other guy, who was me. Marty says 'Sure', so I went on to this set, 300 yards of a recreated Parisian street in Shepperton, complete with traffic and 200 extras, and improvised. The assistant director said, 'Would you like to meet Marty?', so we went into his caravan where he directed via a bank of monitors. It was odd, meeting someone who is very famous, but there is also the banality of him being a little guy with his shirt hanging out."

His most frequent collaborator is Davis, with whom he has worked on Nighty Night, Lizzie & Sarah, Big Train, Jam and Hunderby. "The stuff she does can get so grim, and yet she is the sweetest woman," he says. "Very demure and modest. Where the hell does this stuff come from?" The people who make the weirdest programmes are often the straightest, he says. "People expect Chris Morris to be some cold, satirical ogre, but you could rarely find a more engaging man. There is such great attention to detail. When I did Four Lions, he did years of research on Islam."

Writer Graham Linehan is another sounding board, and was initially involved in It's Kevin. "He was going to be script editor but then got too busy making Count Arthur Strong, but he gave us some early pointers. He made me realise the show should have fewer interviews, less 'And my next guest is' - and more sketches."

He considers himself more an actor than a standup, though he has performed in some of the UK's biggest venues – frequently joining Bailey for arena encores of the hokey-cokey in the style of Kraftwerk ("Bill has already worked the audience into a frenzy. For a lazy man, it's the perfect job").

It isn't always that easy. For his brief scenes with Steve Coogan in I'm Alan Partridge, Eldon spent all day improvising with writers Peter Baynham and Armando Iannucci. "Anything they like, they put in the script – but they also change things right up to filming. At the last minute, Peter would shout, 'Say, "It's like cars, this."' Which makes no sense, but feels like the kind of exchange that might happen with Partridge."

Now it's his turn to take centre stage. "Maybe I wasn't sharp-elbowed enough in the past," he muses. After our interview, I learn that Eldon is a Buddhist, and good at turning negatives into positives. Modesty is part of his makeup. The show opens with him surrounded by lookalike puppets, an experience he says was "rather like being God staring down at his puny creations". He was soon brought back down to Earth. "I showed them to my one-year-old daughter: she was more besotted with the puppets than with me."

• It's Kevin is on BBC2 at 10.30pm on Sunday.

• Bruce Dessau is the editor of comedy website Beyond the Joke

• This article was amended on 14 March to correct the spelling of Graham Linehan's name.