The premise of Fyodor Dostoevsky's 1846 novella The Double is simple but ingenious: a man lives an entirely unremarkable existence until one day his exact doppelganger shows up. This incongruous situation fast becomes insufferable for two reasons: first, the new guy is slick where he is stammering, popular where he's forgettable, Day-Glo to his beige; and, second, because no one else notices any likeness at all between the pair of them.

The Double, it's said, is meant as an allegory: the straight man is Dostoevsky in real life, shy and often awkward; the arriviste is the author 2.0, the person he sometimes wished he was, who is quick-witted and irresistible to women. Dostoevsky believed this was a near-universal fantasy. It was the last book he wrote before being dispatched to a Siberian labour camp.

Richard Ayoade, the comic actor and serious director, wanted to make a film of The Double because it is offbeat and very funny, traits that he shares with the source material. But you can't help wondering if the alter-ego set-up appealed to him, too. In person, the 36-year-old Ayoade is known for being deferential and self-deprecating. He uttered an all-time great line in interviews (which he tends to assiduously avoid) when he told the New York Times: "I'm not a fan of me and I can't become one." The films he makes – first 2010's Submarine, his stylish adaptation of Joe Dunthorne's coming-of-age novel, and now The Double – showcase this side of his personality: smart, finely wrought, inspired by eclectic influences such as David Lynch, Aki Kaurismäki and François Truffaut.

Ayoade the performer, meanwhile, is a more extrovert, populist creature. Standing 6ft 2in, maybe even six-five with the hair, he is someone your eyes always gravitate towards on screen. This is despite the fact that his characters – notably Moss from The IT Crowd or Jamarcus in the 2012 Hollywood caper The Watch – might be self-conscious about the attention. So, does Ayoade share Dostoevsky's belief in a divided self? Do the film-maker and the actor represent twin aspects of his character?

Ayoade considers this suggestion, gazing out of the window of a meeting room at the Imperial War Museum, London, its well-kept lawn and magnificent 15in naval guns glowing in late-afternoon, winter's sun. It's not a conventional location for a celebrity interview, which seems appropriate. Ayoade described his dress sense to GQ as "70s geography teacher" – his style hero is Peter Falk's Columbo – and that nails it: today he wears a brown corduroy suit, floral shirt and thick-rimmed glasses. He proceeds to skirt adroitly around my question.

It really is a masterclass in dissembling. Ayoade starts by saying that any great novelist inhabits all of their characters, and this leads into a favourite quotation from the author Hubert Selby Jr that the ego of the writer should be invisible, while in bad writing it is invariably too obvious. He moves on without pause to Jungian theory and the concept that all stories are essentially varying aspects of one's psyche trying to reconcile into a whole, before concluding: "So I think the short answer would be that I can see Dostoevsky's imagination inhabiting all sorts of things rather than it being dualistic."

I rephrase the question, attempting to elicit a more personal response. This leads into an anecdote that Wallace Shawn, the polymath who appears winningly as a cameo in The Double, told Ayoade about the academic Noam Chomsky. The amazing thing about Chomsky, it turns out, is that whomever he's speaking to – the president of the United States, a class of students – his mode of address and deportment is identical. Ayoade then notes how it's fascinating that you can invariably tell whom your partner or a really close friend is talking to on the telephone, just from their vocal inflections. I realise this will be the closest I get to a private revelation.

This might make our conversation sound frustrating or even awkward; not in the least. It is discursive and surprising in a way these things rarely are. Also, there are signs that Ayoade is slowly reaching his own resolution on the public-private demands of his career. For the longest time, he turned down requests to appear on panel shows and the like, but then last year he took a regular gig on Channel 4's Was It Something I Said? alongside comedians David Mitchell and Micky Flanagan.

What made him change his mind? "It kind of happened by accident," he says, his voice nasal and sardonic. "I imagine after each one that I will never be asked to do another one. I always go, 'Well, that just went so badly. There's no way I'll be asked back.'"

The modesty is reflexive, but then Ayoade corrects himself. "Actually, it was more that after doing Submarine, you do this thing that is completely personal and invested and then you find yourself on T4 trying to see if you can throw a hoop over Justin Bieber's erect nipple and you go, 'I'm in the most absurd kind of weird light-entertainment world anyway.' I remember seeing Michael Haneke doing this Hollywood Reporter roundtable with Judd Apatow and thinking, 'How'd he get there off the back of a film about euthanasia?' And Judd Apatow's going, 'Hey, well…' and cracking jokes.

"So there's no way to avoid it," Ayoade decides. "There's only Salinger and Malick who have any dignity. Everyone else is a gurning buffoon."

Ayoade never especially imagined that he would direct films. "I didn't particularly view it as a possibility at all," he says. "No, I wasn't someone who had a Super-8 camera when he was seven and was making films. I more wanted to be John Squire," – the guitarist for the Stone Roses – "and that proved impossible."

The first film he watched obsessively was Louis Malle's Zazie dans le métro, as a teenager, which led him on to Woody Allen and then Ingmar Bergman, Orson Welles and the people they were indirectly affiliated with, specifically the French new wave. "It's like really liking a band," says Ayoade, "and then you find out who they liked."

Ayoade grew up in Ipswich, studied law at Cambridge and joined Footlights, where he partnered first with The Daily Show's John Oliver and then with Matthew Holness. The latter collaboration would result in the creation of fictional 80s horror writer Garth Marenghi (played by Holness) and his agent Dean Learner (Ayoade). Their second Edinburgh show, Garth Marenghi's Netherhead, won the 2001 Perrier award and was transmuted into a Channel 4 television series, directed by Ayoade. Garth Marenghi's Darkplace was not extensively watched, but had some devoted fans, including Miramax's Harvey Weinstein who wanted to make a movie of it (he would later distribute Submarine in the US).

I ask Ayoade if he's proud of Garth Marenghi, and he practically snorts. "It's a funny thing, you are not proud of things that have anything to do with you in that sense," he says. "I don't know if we made it up or it's apocryphal but we had this thing that Martin Amis's idea of a good time was to smoke a joint and reread his old novels. That to me sums up the idea of being proud of something you've done. A child being proud is fine and even then it's kind of funny: 'Look! I've made this!' Good grief, it's just ewww; it's a bit not right."

How does he feel about The Double then? He spends a few seconds considering his words: "I like it." He contemplates it some more. "Like a lot of things you care about, it's quite private in a funny way for something that's very public. If something's meaningful to you, you tend not to go on about it." Self-promotion also clearly falls into Ayoade's "ewww" category; even the publicity materials for The Double describe it as "a film that resists obvious commercialisation". He acknowledges, "Yeah, that's great for the poster."

David Mitchell, who has known Ayoade for more than 15 years, believes we shouldn't read too much into the deadpan delivery. "He's certainly very self-deprecating," says Mitchell. "I don't think Richard has any fundamental lack of self-confidence but self-deprecation is a good comic point of view to take, whether on a panel show or in real life. And it's much more polite than blowing your own trumpet."

To save Ayoade the embarrassment, The Double is an exceptional work of imagination and storytelling. The American actor Jesse Eisenberg plays the hero, Simon James, and his doppelganger James Simon, and Ayoade drops them in a creepy, dystopian universe where they never glimpse daylight. (It was shot in Crowthorne, Berkshire, on an abandoned development that had a track for testing cars and an underground gun range.) "It's like the future imagined by someone in the past who got it wrong," explains Ayoade. "So the world of the film never existed and never will exist but it has that feeling of a bad utopian dream soured. Like in the 50s, when they thought everyone would have home computers but they would be half the size of a room. It's as if that came to pass."

Craig Roberts and Yasmin Paige in Ayoade's directorial debut, Submarine.

Ayoade's direction is assured and meticulous; he spent five months on the sound alone, creating just slightly off-kilter effects to match the film's aesthetic. He might have envisaged that his experience on Submarine would make some aspects of the production more straightforward, but that turned out not to be the case. Thus it was that a film whose first draft was completed in 2008 took more than five years to finish.

"It's a completely gut-churning experience but it's really exhilarating at well," says Ayoade, who co-wrote the screenplay with Avi Korine, Harmony's brother. "And it's sort of continually terrifying because it's just always evaporating. I heard Stanley Kubrick describe it in an interview as one of those games where you are trying to get all the balls in the hole. If you spoke to someone playing that game and said, 'Are you enjoying that?' He'd go, 'What are you talking about? I just need to get the balls in the hole!' You reach a point where any enjoyment has gone and you just have to do it."

Those who are hankering to see more of Ayoade in front of the camera might face a lengthy wait. In fact, he doesn't even really consider himself an actor. "Just because you're in something doesn't mean you're an actor," he clarifies. "I have a limited range and have worked largely for friends. It's just not something I particularly think of myself as having a skill-set at."

It's likely he will reappear in new series of Was It Something I Said? and Channel 4's tech show Gadget Man, on which he replaced Stephen Fry as presenter last year. He would also like to make more films – if anyone will let him. Which, naturally, he considers improbable. This time, however, he's not just being modest. He needs The Double to do at-least-not-disastrously at the box office, which is why he's prepared to endure the torment of doing interviews to promote it.

"It's so inconceivable that you could get to make a film," he says, as night draws in outside. "It's so difficult to imagine even until the very day that you start that it's going to happen. They are always collapsing and it's so hard to get everyone together and it's such an amazing thing that you could get funding to do it. I always think Boondock Saints," – the 1999 film starring Willem Dafoe that bombed at the box office (though it had a second life on DVD) – "when someone announces their plan: 'It's going to be the greatest thing ever and I'm going to make four films…'"

Ayoade crosses his legs, uncrosses them again. "To me it sounds like people announcing their plans at the start of The Apprentice. Yep, well, good luck with the expansion! I don't know, who knows?"

The Double is out on 4 April