Ben Whishaw has just walked into the Jerwood Space – where he has been rehearsing. Slight and boyish, he is wearing a shirt and T-shirt in jostling colours – plum reds and pinks – as if he has grabbed the first clothes he could find. He certainly does not look as if he has spent hours communing with his mirror. Like Hamlet – the part in which, at 23, he made his name in Trevor Nunn's production – there is distraction in his aspect. When he talks, he has a way of tugging at his hair – as if trying to pull it out. There is a sweetness in him too – evident straight away. He gives the impression of being an accidental pin-up, bemused by the way people see and often – probably inconveniently– fall in love with him. You can see why he was cast as Keats in Jane Campion's film Bright Star (though better looking – Keats's face was more irregular). Now, at 32, he is about to open in John Logan's new play Peter and Alice as Peter Llewelyn Davies who lent his name to Peter Pan. Actually, though, what marks Whishaw out as an actor more than any lost-boy glamour is his voice. His fastidious diction is unmistakable: he picks up words as if with sugar tongs – as if each syllable needed personal attention.

I begin by telling him I am in close touch with his fan club (at least, a splinter group: my niece and her friends). I don't tell him that, snug in my pocket, is a text relating how one of my niece's friends recently saw him in the street and was so overcome that she walked straight past him, then had second thoughts and turned back. The words that came out of her mouth were not the ones she had planned. She shouted: "I love your coat!"

I have to confess to having been more anxious than excited at the prospect of meeting Whishaw, having read one piece too many pronouncing him uninterviewable, shy and liable even to treat questions about his roles as booby traps. Yet now, when I suggest he hates being "interviewed" and commiserate, he laughs and says: "I don't." It seems a good start.

Judi Dench and Ben Whishaw in John Logan's Peter and Alice: 'What I love about Judi is her playfulness and lightness.' Photograph: Johan Persson

And in this context, it is fitting that Peter and Alice should be about literary identity theft: an exploration of the unhappy notoriety of being written about as – and overtaken by – a fictional character. Alice Liddell Hargreaves was the inspiration for Lewis Carroll's Alice in Wonderland. It was she who asked for the story that was first told in Oxford, in a rowing boat on the Isis, en route to a picnic. Peter Llewelyn Davies was the middle son of Arthur and Sylvia, one of the boys informally adopted by JM Barrie. And although Peter Pan's character was modelled more on his brothers, George and Michael, Peter was – because of his name – never to shake off the Peter Pan tag (he referred to the book as "that terrible masterpiece"). He committed suicide in 1960.

It was The Real Alice, the biography by Anne Clark, that alerted John Logan to the occasion upon which Peter Llwelyn Davies and Alice Liddell Hargreaves coincided, in London, in the quaintly named Bumpus bookshop, on 26 June 1932. He "wondered what they said to each other…", and his play started to take shape. The result is Alice – and Peter – through a glass darkly. Judi Dench plays Alice; Michael Grandage directs his company's second production in its year-long season at the Noël Coward theatre.

John Logan was also responsible for the screenplay of Skyfall, the latest Bond film, in which Judi Dench and Ben Whishaw starred as M and Q. On set, Logan asked: "Have you read my play yet?" And when Whishaw eventually did, he was "moved" because: "I didn't know their story or how desperately sad their lives had been." Up to that point, his acquaintance with Peter Pan had been jocular, courtesy of Disney and the local pantomime in Langford, the Bedfordshire village where he grew up (he played John). It is only recently that he has begun to see there might be something sinister about the boy who would not grow up. "Heartless" was one of JM Barrie's favourite words.

"There is a conflict between the honouring of a living person and of a writer's work," says Whishaw. He is an avid reader and has found "a lot available to read about Peter Llewelyn Davies". Inevitably, he has made discoveries that go beyond the play. At the moment he is engrossed in Pat Barker's novel Regeneration which describes the effects of shell shock in the first world war. Llewelyn Davies returned shell-shocked from the front and went to an asylum: "He lost his mind… his brothers said he never really recovered." He describes Peter as "famously – or infamously – melancholic, yet witty too". That must be interesting to play? "Yes – but we are on our first run-through, so I don't know what will come out of it."

I have a theory about Whishaw's desire not to commit himself. It seems to go beyond any commonplace need for privacy – it is as if he wanted to keep himself private from himself – as if he feared over-analysis might scare talent away? "I always feel I am in the dark. You are never finished… it is not as if you can look back and think: ah… I know what I am talking about. You are only as good as your last job and are always struggling and striving and you never quite get to where you want to be…"

And then we talk about acting as a promiscuous profession – the falling in love with roles, the moving on: "It can be a very strange feeling. I was in a bar having a drink the other day when this guy came up to me who was some sort of healer. I said I was an actor. And he said: 'Ooh, I am very concerned about actors – you shouldn't be going through those things because the body doesn't know when something is real and when something isn't.'"

This makes sense to Whishaw. And, ironically, it is something John Logan's play also acknowledges: "the hazy line between fiction and you". Once a role is over, Whishaw forgets it: "Like Peter Pan, who says: 'I forget them after I kill them.' You can't linger." But he does regret that the BBC2 1950s newsroom drama, The Hour, by Abi Morgan – in which he played Freddie Lyon opposite Romola Garai – did not make a third series. It is easier to look back at Q in Skyfall because there is more to come. No computer boffin himself, he is hopeful – basing this on "tiny teasers" from John Logan – that in the next film he will be released from "that horrible bunker" and "go wherever the action is happening".

Whishaw also stars in Lana and Andy Wachowski and Tom Tykwer's new film of David Mitchell's byzantine novel Cloud Atlas, alongside Tom Hanks, Halle Berry and Hugh Grant. He carries off five wildly different parts – the most striking of these being the dishy homosexual composer Robert Frobisher, hoping to compose a 20th-century masterpiece. But he also plays a cabin boy, store clerk, tribesman and a demure wife, complete with blond ringlets, to Denholme Cavendish (Hugh Grant). Together, we alight on the interesting truth that the film's central statement "Our lives are not our own" nicely defines acting itself.

It is a film determined to make you think – about free will, for example. Is Whishaw in the mood for a spot of philosophy? Where does he stand on this one? "Everything is a choice. Your one certainty in life, your power as a human being, is that you have a choice in every situation about what you do next and about how you take what has happened to you." He loves, too, Cloud Atlas's wacky version of reincarnation. "I've recently been to Sri Lanka and loved the Buddhist idea. It is lovely to think that our time as humans is a preparation and that we have evolved from some lower life form [he giggles]." And what would he like to come back as? "I'd happily come back as some animal." Which one? He hesitates: "I reckon domesticated cats have a pretty good life."

At the same moment that Ben Whishaw was evolving into a human being, his brother was too – he is a twin. His brother is "6ft and fair and doesn't act". His mother is a non-identical twin also: "So that was our reality." She worked for John Lewis. His father played football for Stevenage before moving into a career in IT. Recently, his mother told him that, when given identical presents to his twin, he would protest: "'I'm not interested in toy cars – why have I been given this?'" Yet he adds: "I don't remember ever feeling resentful about sharing a birthday."

Would he have preferred to be an identical twin? "My instinct is that I wouldn't have wanted that for myself." As a schoolboy his interests included Hitchin Youth Theatre and abstract painting: "I would start something and tell myself: I have to make a painting out of this in the next two hours… where does it lead me?" It led him, briefly, to art college but he "scrapped" it when the pull of theatre showed no sign of loosening – and got a place at Rada: "I value so much the friendships I made there – not just with people in my year but with tutors." And London was a blast for a boy "from the sticks".

He describes himself as a level character: "I am a typical Libran. I tend to see two sides of everything." He is far less "paralysed" about performances than he once was. He believes Simon Russell Beale had it right when he said all actors have to fear is "vanity – the fear of it being hurt. So the thing is to know your lines – then do it. Bringing it down to something simple and practical is helpful". He likes to think of himself, especially in the theatre, as "a channel for other people to feel – for, in a sense, it isn't about you".

And then we start to talk about what makes great acting (as opposed to merely "good"). Whishaw says he has just been moving house: "While I was clearing I came upon, in a big old box of rubbish, a VHS cassette my grandad made for me in 1995 of a South Bank Show about Judi Dench." He remembers watching it and its "magical" effect precisely because: "I couldn't see how she was doing it." And now there is the uncommon satisfaction of acting alongside her: "What I love most about Judi is her playfulness and lightness. You never feel anguished – as you sometimes can in rehearsals when you feel you are never going to get there. She is such a warm, loving person – but in an unforced way. Same with her acting – nothing effortful."

He was also bowled over, recently, by Harriet Walter in Phyllida Lloyd's all-female Julius Caesar. "Who wouldn't want to see Harriet Walter play those male roles? She was incredible playing a man." Whishaw's own great Shakespearean ambition is to play a woman: Viola. He told Julie Taymor this (he was Ariel in her film of The Tempest) to which she has apparently replied: "Well, we'll do that."

Whishaw's other great hero, as a teenager, was Michael Gambon (who played his father in Brideshead Revisited, the 2008 film directed by Julian Jarrold). What do the great actors have in common? "It could only ever be Michael Gambon doing the part that way. He has some access to something that is totally him. With all the great people, they have a particular idiosyncratic quality that means they are always different and always the same."

It is incredibly nice to chat with Whishaw – the conversation flows. Against expectation, he is such easy company. I privately reflect that the "particular idiosyncratic quality" he mentions could apply to himself. Does he like talking about what makes other people tick? "It is my favourite thing in the world. Who doesn't love that? I don't think I am especially interested in celebrities but I love talking about what is going on with people and why they do what they do." And then I ask what he does to relax, whether painting still absorbs him, and he volunteers that photography has taken him over: "I am obsessed. It is my new thing and it is really addictive." He launches into an enthusiastic spiel about "forgotten photographs by Vivien Maier of 1950s and 1960s New York". And he explains: "I have been buying up old film cameras. The oldest is probably from 1980." It is the unpredictability Whishaw prizes most: "I hate pseudo-perfection, that horrible crispness where there is no magic." It takes one straight back to what he said earlier about the need, as an actor, to stay in the dark.

Peter and Alice is in preview now at the Noel Coward theatre, London WC2, opens on 25 March and runs until 1 June