The first time Kristen Schaal performed in Edinburgh, she became famous in New York.

Go figure, as they say over there. The stateside acclaim wasn’t for her 2007 Fringe show, however, though it was well-enough received. The real reason was the simultaneous television premiere of Flight Of The Conchords, the HBO comedy series in which she featured. Schaal played Mel, a married woman obsessed with the struggling Kiwi folk duo of the title (also known as comedians Jemaine Clement and Bret McKenzie).

“I remember going to an internet cafe in Edinburgh – I was on MySpace at the time – and all of a sudden I had 100 new friends,” she says. “I thought ‘What’s going on?’ I started getting the feeling people were watching the show. Then I flew back right into it. It was like ‘Oh my God, people are screaming at me!’. It was weird. It was a unique and peculiar experience to have.”

Until then, the 33-year-old had worked almost exclusively in that corner of the New York live scene where vaudeville, stand-up and improvisational theatre meet. She still does, in fact, though when she arrives in Edinburgh next month for a fourth stint at the world’s biggest arts festival, she brings with her a growing reputation as a quirky screen actress. She may not yet qualify for top billing alongside female comedians such as 30 Rock star Tina Fey and namesake Kristen Wiig (of Bridesmaids fame), but you sense it’s only a matter of time before the bit-parts and cameos turn into something more concrete.

You see, what Schaal has besides quirkiness is a deep and abiding likeability, a key quality when there are audiences to win over and commissioning editors to impress. Perhaps it’s the Shirley Temple hairdo that does it. Perhaps it’s the voice, which can turn from helium light to foghorn loud in the time it takes to draw breath, or the material which goes from innocent to filthy at the same speed. Perhaps it’s her performances in film and TV favourites such as Toy Story 3 (she voices Trixie the Triceratops), satirical US import The Daily Show (she guests as Senior Women’s Issues Commentator) or animated series Bob’s Burgers, currently showing on Channel 4.

Or perhaps it’s simply because, as one comment beneath a YouTube clip of her puts it, she looks like “a hot alien”.

“Ha ha ha ha ha,” she bellows when I put this last one to her. “I’ll take that!”

Schaal was born and raised in Colorado. Hers was a rural childhood – her parents had a farm – and she got the comedy bug in high school when a song she didn’t think was funny had her classmates in stitches. “I don’t think it was necessarily a humorous piece,” she deadpans. “But I decided to explore that.”

After university, the exploration continued. It brought her to New York, to the Brooklyn stand-up scene, and to a waitressing job which she stuck with even after 2005, the year she made New York Magazine’s Ten Funniest New Yorkers You’ve Never Heard Of list. Along the way, she also discovered the joys of improvisational theatre – or “improv” as it’s known.

“It becomes this addiction,” she says. “As soon as you start it, that’s all you want to do. You want to perform over and over. It’s almost like cracking a riddle – you learn how to perform a scene out of thin air and write a story from nothing. And I think that’s what’s so exciting about improv. It challenges your imagination at the same time as it teaches you about storytelling.”

It also stands its best practitioners in good stead when it comes to screen roles, as Schaal is discovering. “Right now, improv is hot. Film directors want improvisers. They want you to say the lines but they want you to go off-script too. They encourage it. After you’ve done several takes, they’ll even go so far as to say: ‘OK, this one’s off-script. Go!’ So it’s become a real trend in the industry.

“There was a lot of improv in Flight Of The Conchords. If I wanted to add a line in, I’d add a line in. But at the same time I always want to be careful about how much is improvised in shows. I might improvise an extra joke but I’m not going to improvise the structure of the scene. The ultimate joke of the scene is something the writers are responsible for and deserve the credit for.”

It was through working at the People’s Improv Theater in New York that Schaal met Kurt Braunohler. They bonded over what she calls “an affection for absurdity and surrealism”. He’s now her regular on-stage collaborator and, for the last six years, her partner in Hot Tub, the show she’s bringing to Edinburgh. Her boyfriend, meanwhile, is Daily Show writer Rich Blomquist with whom she wrote The Sexy Book Of Sexy Sex, a spoof instruction manual published last year. It’s about sex.

Like everyone else, Braunohler is nuts about Schaal. “She’s like a giant bag of kittens that have organised themselves to walk around like a human,” he tells me. “She’s playful and adorable and she drinks milk out of a saucer.”

Hot Tub is an award-winning Monday night variety show the pair host in Brooklyn. They charge a nominal entry fee of $5, cheap by the standards of most New York comedy clubs, and use the slot to give stage-time to up-and-coming acts and, importantly, to hone their own material. Schaal describes it as “the springboard” for pretty much everything she goes on to perform elsewhere.

“If you’re charging someone $5 to see a show, I have no qualms about trying out untested material that might completely bomb,” she says. “If I’m getting paid a lot of money to do a show, I’m not going to take any chances. I’m going to bring out the A-game because I owe it to the audience. But you can’t get to that A-game material unless you start somewhere.”

Edinburgh, rest assured, will be enjoying the A-game in a souped-up version of Hot Tub which will feature three different guests each night (Peep Show’s Isy Suttie has already been pencilled in), a house band (New York three-piece Adira Amran And The Experience) and, of course, Schaal and Braunohler’s absurd take on things.

She may be a bigger name now than in 2007, but the Fringe is still the Fringe. Having performed to 17,000 people at the Hollywood Bowl with the Flight Of The Conchords live show, Schaal will perform in Edinburgh at the significantly smaller George Square Theatre as part of the Assembly programme. This, though, is a good thing. “You can’t see anything on those big stages. It’s like a giant, black, gaping mouth. You have to throw jokes into the giant’s mouth and hope it likes them.”

When the Fringe finishes, it’s back to New York where she’s in discussions over a film role she’ll say no more about now. Next, we’ll see her play alongside Hugh Jackman in Butter, a comedy about the competitive world of butter carving and hear her in animated TV series Gravity Falls.

But if anyone has any sense they’ll seek to team her up with Russell Brand again – that “hot alien” crack came attached to a clip of Schaal and Brand ad-libbing uproariously in the Briton’s trailer during the filming of Get Him To The Greek. Schaal played a news correspondent in the 2010 comedy.

“I find him incredibly witty and quick and all of the times I’ve met him he’s been sincerely nice,” she says. “A really good guy. I think he’ll have a long career in America”.

Well, only as long as he keeps his mouth in check when leaving messages on answer-phones, I venture. “I know all about that,” she laughs. “But he can’t go too far over here. I’d like to see him try.” With Schaal as a big-screen sidekick, he just might.

Hot Tub With Kurt And Kristen is at Assembly George Square (August 12-27). For tickets visit www.edinburghsbest