What Murray did next: Kiwi stand-up comic Rhys Darby steps out of the shadows of Flight of the Conchords

He stars in a hit TV show, upstaged Jim Carrey in his silver-screen debut and has just completed a sold-out series of stand-up gigs in London. So why is it Rhys Darby still only gets recognised once he's opened his mouth?

Generally when I open my mouth, people recognise my voice. I normally wear a hat and glasses, not for a disguise, just for..." Your trademark look? "Yeah! It sort of became my thing." Fame has not come easily to Rhys Darby. Indeed, the name might not even mean much to you now – but take a quick look at that picture to the right. Ring any bells? A little bit. A little bit, eh?

That line, delivered as something of a catchphrase in the 35-year-old New Zealander's distinctive, slightly nasal intonation, would surely be the giveaway to anyone who has caught his turn as the well-meaning but inept band manager Murray Hewitt on cult HBO comedy Flight of the Conchords (shown here on BBC4). But while his fellow Conchorders Bret McKenzie and Jemaine Clement became instantly recognisable (thanks in no small part to some prodigious facial hair), Darby left the set of the first series to go back to his first love of stand-up, performing in tiny gigs in student pubs in Wales, because "I didn't know Conchords was going to be a success. We thought it was a bit too novelty for it to be as big as it's got." How big? Big enough to catch the eye of the Hollywood director who gave Darby his first taste of the silver screen – but we are getting ahead of ourselves.

Darby's inability to predict the success of the show may well stem from a character trait he says is germane to his countrymen, of not believing in themselves ("It comes from living on a small island in the South Pacific, the furthest away from anyone, and us looking to the rest of the world rather than anyone looking in") – a trait that meant Darby had to struggle in his early years just to get up on stage. "No one in New Zealand did stand-up until 15 years ago," he reveals. "We didn't get our own purpose-built comedy club until 1996 – and we've still only got the one, in Auckland."

Indeed, the greats of New Zealand stand-up are far from household names here – and even struggled to get by on their funny bones at home. Billy T James, after whom the country's most prestigious comedy award is named, was an iconic Maori entertainer, but he had to rely on his all-round repertoire of singin', dancin' and paintin' talents to be given a much-loved TV show in the 1980s (Hang on: paintin'? "Yeah, he was a bit like Rolf Harris," explains Darby) – and his fame travelled no further than Australia. Similarly, John Clarke, whose portrayal of country farmer Fred Dagg endeared him to the nation in the early 1970s, left for Oz at the height of his fame "because New Zealand didn't really know what to do with him; we were too conservative to give him his own show".

Lacking a heritage into which he could tap, Darby helped found a stand-up scene in Christchurch while at university. "I was part of a troupe called Four Fingers Missing. We'd go to local pubs and ask if we could perform. We weren't asking for money, just a chance to try out our stuff. And the reaction back then was, 'Who do you think you are? Here we go, we've got some Billy Connollys here.' New Zealanders just didn't think we [as a people] were capable of doing anything like that. Of course, we went on stage, did our stories, some surreal stuff, some songs, and people started to say, 'Ah, you're not too bad, actually... we'll buy you a beer.'"

The scene grew to the extent that New Zealand now hosts an international comedy festival every year. "And we've got at least five or six comics now!" The global triumph of Conchords has helped in no small part. "Although we had those few home-grown comedy heroes like Billy T, it was only after crossing the border and having the rest of the world go, 'Oh my God, you guys are funny,' that New Zealand went, 'OK, we'll accept it now, maybe we can do it.'"

This shifting sense of confidence has itself proven a rich seam that Darby has mined in both his stand-up and his screen roles. His "Imagine That!" stand-up tour of last year was filled with his own tacit acknowledgement of his talents followed immediately by self-deprecation ("Thanks for the applause on the helicopter [impersonation] there. Do you need friends when you can do that...? You do. I used to do it when I was a kid and my friends just thought I was a dick,") while Conchords' Murray Hewitt is a character who "has this new-found confidence, but with all these back-of-the-head insecurities" – his heartbreaking ode to the "leggy blonde" using the office photocopier during the second series of Conchords being a case in point. His first big-screen role, as Jim Carrey's fancy-dress-obsessed boss in Yes Man was similarly endearingly nerdy.

Yet Darby has since moved on from playing hapless managers, first appearing alongside such thespians as Philip Seymour Hoffman and Kenneth Branagh in Richard Curtis's The Boat that Rocked as a DJ – "though my character was still very losery, even more of a loser, really; he's thrown off the boat" – and next year he will break the mould entirely as the lead in the romantic comedy Coming & Going, in which he plays "a hunky doctor - HA!" opposite Sasha Alexander, his Yes Man co-star. "He's definitely not a nerd," insists Darby. "But he is still zany."

Those movie roles play a large part in a short stand-up run Darby has just completed at the Bloomsbury Theatre in London and which he is now taking to the Edinburgh Fringe Festival, in which he relates his journey from small-town Kiwi boy to Tinseltown leading man. "For the most part, since I've become famous – it always sounds awful when you say that – it's an adjustment, and I wanted to laugh with people about those changes. Two years ago I was performing in tiny pubs in Wales, and suddenly I get this phone call from a Hollywood director saying that Jim Carrey wants to work with me. Jim Carrey!

"After the first season of Conchords, I was really at a loss as to what to do, because I'd gone to America and thought, 'Here we go, I made the big move, I've given up the stand-up, it's going to be all acting from here,' and... nothing happened. The show hadn't gone on air yet, and there's no money doing stand-up in America unless you're a big name. Then Conchords was on HBO, word spread and it gained a big audience – including Peyton Reed [the director of Yes Man]. I was just lucky. What are the chances of your first big movie being with Jim Carrey? I'd idolised him."

Carrey himself likened the New Zealander to Peter Sellers – quite a claim. "We had some fun improvising stuff," Darby explains. "I improvised a lot more than he did – quite often because I couldn't remember the script. Half the time it's funnier than what they've written anyway. I'd look over to the writers and they'd put their thumbs up as if to say, 'That's awesome, we'll take credit for that!'"

It was less Peter Sellers than his erstwhile Pink Panther aide Cato that a frenzied Darby brought to mind in an advertisement he filmed last year with Roger Federer for Nike, playing a crazed fan who believed he was the Wimbledon champion's coach. The part, for which he leapt about the tennis star's living-room in a duel of thrumming forehands and sizzling smashes, took advantage of the physicality that has been a focal point of his comedy career. Whether it's his energetic impersonations of robots, jetpacks or dinosaurs, or bouncing around the stage to demonstrate the innate hellishness that is sitting in a stationary bumper car, Darby has been throwing himself around stages for more than a decade – and to ever-increasing crowds.

"Before I left New Zealand, I was doing more gigs than anyone in Auckland – three a week. And half the time the audience would be the same people. At the one club, it would be 60 to 100 people a night, and the other two gigs would be 20 to 30. So when I first came to Edinburgh, in 2001, in a show called The New Zealand Brat Pack – we wore suits and drank Martinis – I was blown away. This amazing city, full of entertainers, 200,000 people. I fell in love with it and vowed to come back every year no matter what."

Which he did, for five years. Now, after two years away, Edinburgh is where Darby finds himself again. And chances are, this time one or two people (let's not get over-confident; it's not the Kiwi way) might even recognise him before he starts gabbing away.

Rhys Darby is performing at Udderbelly in Edinburgh (020 7222 3311) until 15 August. The second season of 'Flight of the Conchords' is out on DVD tomorrow

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John Clarke

"A huge influence," says Darby. "Everyone could relate to his farmer character Fred Dagg." After moving to Australia in the 1970s, Clarke wrote satires and screenplays, including one that became the 2001 Billy Connolly vehicle The Man Who Sued God.

Billy T James

After performing with the Maori Volcanics Showband in the 1970s, James went solo in the 1980s, doing sketches and impressions and singing cabaret.

McPhail & Gadsby

"They were a duo who did political satire in the 1980s. They were very much stamped on our consciousness."

Ben Hurley

"He's come over here the past three years. He does a bit of politics, he's a bit like [the Canadian comic] Glenn Wool, a bit provocative, but not too bad." Won the Billy T award in 2004.

Sam Wills

"He's more your physical street performance-based maniacal idiocy comic. He did a show called 'The Boy with Tape on his Face'. He just had tape on his face and it was all mime. He does crazy tricks, too: he swallows swords, puts his body through a coat hanger."

Rhys Mathewson

"He's a very young comic; he's got the same first name as me, so I'm a big fan. His style is story-telling and he also has that confident-naïve guy mentality. I think he's going to make waves."