Eddie Redmayne: The darling of the Donmar is making tracks into Hollywood

Eddie Redmayne is, by common consent, one of the most exciting actors to hit both stage and screen since the auspicious advent of Mark Rylance some 30 years ago. Indeed, it was Rylance who gave Redmayne his first break, casting him as Viola to his own astonishing Olivia in the all-male Twelfth Night that the Globe performed at Middle Temple Inn in February 2002. This staging was designed to mark the 400th anniversary of the occasion when Shakespeare's men crossed the Thames to perform the self-same comedy for the lawyers.

As he was still an undergraduate, reading history of art at Trinity College, Cambridge, Redmayne was not able to perform the role again when the production moved to the Bankside theatre the following summer. He had prior engagement with his finals. I was lucky enough to be invited to the Middle Temple shindig and immediately registered that here was the first sighting of an extraordinary talent. But my review of his performance got comically garbled. Of his full-lipped, highly strung colt of a Viola, I wrote that s/he "would bring out the bisexual in X" and at this point I named a famously pious, repressed and repressive living personage. It was meant to be a cheeky way of saying that Redmayne's cross-dressed performance would sexually disconcert the locked-up likes of the puritan steward, Malvolio. But, with the prospect of possible litigation, the sentence was both reduced and inflated to "bring out the bisexual in any man" which, with its grandiose generalisation, managed to turn an intended garland into albatross.

Lanky, glittery-eyed, and blessed with the ability to move in a manner that seems to grant unimpeded access to the friction of the inner life, the erstwhile model for Burberry has several times been cast as a gay young man with a father-complex, but he's also a powerful draw to women. When I met him in his dressing room at the Donmar, where he has won rave reviews as the assistant to the artist Mark Rothko in a new play Red, I told him of my encounter with the female assistant at the Leicester Square HMV during my purchase of the DVD of Savage Grace. This is the 2007 Tom Kalin film about the hideous failure of will that befalls the beneficiaries of a load of inherited loot. All drug-zonked, underlying anguished lassitude, Redmayne brilliantly plays the predominantly gay, American scion of a moneyed union where heat meets cold leaving the son as the resulting steam. The HMV assistant told me that it was a great film, and I told her that I was about to interview "the boy". Her ears pricked up. "Well, you tell him 'well done' from me, and please give him my address," she declared.

Redmayne slaps his thigh with amusement at this story. He's unassuming, very smart, and humorous. Now 28, he grew up in a household that "did not have the theatre gene" – his father is something big in the city; his mother used to work in relocation. His brothers are all sportsmen and he cheerfully reports how sometimes he mentally runs roles past their tastes and preferences. "They once begged me never to hit them with a Restoration comedy". Seeing him play the MC in PVC and fishnets in Cabaret at the Edinburgh Festival when he was 19 (it's a role he hankers to portray professionally one day) persuaded his parents that maybe he could make a career in the alien world of theatre.

On stage, he has the knack of embodying people who seem to have a layer of skin missing and an extra mental radio station picking up frequencies unheard by other characters. Offstage, in contrast, he comes across as a man who is at home in his own regulation-strength skin and who, rather than a seer's intense tunnel vision, has a canny, wittily expressed, circumspection about life and career. He's also disarmingly honest, particularly with regard to the vagaries of movie making.In Savage Grace the parents were played, magnificently, by Stephen Dillane and Julianne Moore who are, in some ways, chalk and cheese. "Stephen is incredibly cerebral and has a meticulous way of rehearsing. Julianne is a shooter from the hip – she doesn't like to go straight into scene. And as the kid, I was sitting there thinking, 'Oh my God these are two hugely different ways of working'. But that was wonderful for capturing the character of a boy who is pulled in two directions..."

His Eton education did give him a leg-up, he's only too willing to acknowledge. Simon Dormandy, a former actor, became head of drama there and instilled a professional discipline in matters such as verse-speaking in his theatrically inclined charges. Among other roles, Redmayne played Shakespeare's Henry VI and his first drag turn as Adela Quested in a stage version of A Passage to India. Siobhan Bracke, the casting director, knew Dormandy from RSC days, and approached him when the decision was made to field a very young man as Viola in Twelfth Night. By this stage at Cambridge, Redmayne got his second call from Rylance and director Tim Carroll, "when I was well into a bottle of wine with a friend in Notting Hill. I had to go straight to the Globe and do a scene with Mark who I think was in a rehearsal dress and the wine..."

On screen, "I seem to have appeared in anything with Elizabeth I in it", he laughs, referring to the HBO mini-series Elizabeth I. He's also popped up in period pieces The Other Boleyn Girl and 2008's BBC triumph, Tess of the d'Urbervilles, in which he offered a masterly study in the magnetic properties of anguished self-deception as Angel Clare. On stage, he has now played three young American men who have issues with their sexuality and their father. In 2004, in a performance which won him the Evening Standard Award for Outstanding Newcomer and the Critics' Circle prize, he had to fall into a needy kiss with Jonathan Pryce who portrayed the bestiality-convert father in Edward Albee's The Goat. Preppy and peppery, he brought a lovely wounded flounce to the son of the US President-to-be in the excellent election-night political play Now or Later at the Royal Court in 2008. And now in Red, excellently directed by Michael Grandage at the Donmar, he plays Ken, a fictional composite of all the assistants ever used by the intense, dogmatic Mark Rothko. It's again a play about a father-and-son relationship in which the younger generation begins to make its superseding force felt.

Grandage says that actors often divide into those who are good on the front-foot and those who specialise in the power of passivity: "Eddie is terrific at both. You've only got to look at the change in the way he moves in the play. His feet are barely touching the floor in trepidation at the beginning and later he's virtually swaggering to the gramophone when he starts playing jazz records."

The actor's powers of observation are remarkable. It's a gift likely to be evident in his new film, The Yellow Handkerchief, with William Hurt and Kristen Stewart, which opens soon in the US. "It's a road movie and I play an adopted Native American who maybe has Asperger's." Redmayne imitates the way the character has little sense of the fly-zone of other people's personal space.

As for his future on the stage, he is surely now in a position to start calling the shots. I tell him that he will be on a ration of two-star reviews in The Independent for all stage work now until he accepts his destiny and plays Shakespeare's Richard II. Top directors should be forming an orderly queue of supplication for his services in this. While I was making ready to depart, he informed me that ironically, given his current role in Red and his art history background, he is in fact colour blind. I jocularly accused him of bluffing in the interests of publicising the show. But I rather think that Redmayne isn't programmed for fibbing offstage, just as onstage it's impossible to think of him making a sound or gesture that isn't full of truth.

'Red' to 6 Feb, Donmar Warehouse, London WC2 (0844 871 7624); 'The Yellow Handkerchief' is released later this year