The day before I'm due to meet Daniel Radcliffe, I witness what can only be described as scenes of absolute hysteria.

At the Venice Film Festival, outside a beachside bar where the former Harry Potter star has been holding court to publicise his new film Kill Your Darlings, the entrance is swamped by hundreds of teenage girls, screaming uncontrollably. As burly security guards separate the wailing throng, Radcliffe, at 5ft 5in not the tallest man in the world, makes it into his chauffeur-driven car - just. Beatlemania had nothing on this.

Thankfully, the next day when we meet in that same bar - Radcliffe wearing a white shirt, camouflage-print trousers and a turquoise jacket - he's still intact. "It's lovely that people are still so effusive, to say the least," he grins. "Five hundred people chased me to the toilet yesterday! The toilet is 300 yards away from here - maybe not even that - and Sam, my security guard, was like, 'We're going to be running, mate, just so you know.' I made it, thankfully, but people were trying to come in in front of me. I was like, 'You can't come into the toilet with me!'"

Bathroom bothering aside, the 24-year-old - Dan to one and all - is at ease with all this. "You have to find it funny. If you take it seriously, you'll either be frightened or arrogant, because it is overwhelming. You have to put it into perspective: anyone who has been cast in Harry Potter would now be getting this." That's modest of him; it's hard to imagine his co-star Rupert Grint - who played Ron Weasley and is currently on stage in London in Mojo - attracts quite the same crowds at the stage door.

It's been more than two years since Radcliffe ceased playing Harry Potter in an 11-year odyssey that stretched across eight films and his entire teens. With the franchise grossing $7.7 billion, it made him - according to the Sunday Times Rich List - the wealthiest British actor under 25, with an estimated fortune of £60 million. Those who know him say it hasn't changed him a bit. "He's the sweetest, nicest multi-millionaire I've ever met," laughs actor Peter Mullan, who worked with him on the final Harry Potter film.

Radcliffe says that what has changed is the heat around him. "It's become a lot more intense," he says. Which feels almost perverse. It's not as if Radcliffe has courted fame. He's deliberately avoided other franchises or blockbusters - even if the 2012 film version of The Woman In Black, in which he starred, became one when the modest British chiller raked in $127m. Instead, he's chosen to tackle challenging roles on stage. After making his Broadway debut in the musical How To Succeed In Business in 2011, this year in London's West End he played a good-natured Irish orphan with dreams of stardom in The Cripple Of Inishmaan.

Even mid-Potter, Radcliffe took a big roll of the dice, starring in a 2007 production of Peter Shaffer's play Equus as the disturbed stable boy Alan Strang. Every night, it meant stripping naked - with nowhere to hide. "Doing Equus was a big thing," he nods. "If you let it be known to the world that you want to take risks and push yourself, people will give you opportunities. Whether you make the best of those opportunities is up to you. But as a statement of intent, Equus was very important in this whole transition."

You might say the same for Kill Your Darlings, which is unquestionably his most adult film role. Playing his first real-life character, he stars as a young Allen Ginsberg, the iconic gay Beat poet and author of the controversial Howl. Set in 1944, when Ginsberg was a student in New York, we see him long before he began to look and sound the way he's remembered. "He hasn't done a ton of drugs at this stage in his life, nor has he smoked for 20 years," notes Radcliffe. "So it's not the voice that probably everyone knows as Allen Ginsberg."

Without the famous Ginsberg bushy beard, Radcliffe's version is on the brink of adulthood, as he's introduced to the likes of Jack Kerouac (Jack Huston) and William Burroughs (Ben Foster). "I think what will be exciting for people is seeing these three guys before they were the venerable old men of American literature," says Radcliffe. "They were just running around New York, tearing it up, doing drugs and having this wild life, and becoming liberated. It's the story of that time in their lives and it's the story of them all meeting."

Raised in Fulham, Radcliffe's background has a literary slant too. His mother Marcie is a casting agent while his father Alan gave up his job as a literary agent to chaperone his son after he was chosen from thousands to play Harry Potter. "When I was at school, I wasn't particularly good," says Radcliffe, recounting his time at Sussex House in Chelsea. "I found it hard to concentrate."

Yet receiving private tuition during the Potter years brought the scholar out in him. If he wasn't an actor you could imagine him as a librarian or literature academic.

"I love poetry and I always have," he says. "I definitely didn't have the same period of connection with the Beats that a lot of people have. But I respected what they stood for. But I like all the verse that Allen was railing against - form, metre, rhyme, I love all that. I've always enjoyed poetry. Some poetry is very dense but a lot of it's really worth it for me. When I read [John Keats's] Ode To A Nightingale for the first time, I didn't understand all of it or what it was saying, but with a little investigation it becomes even more beautiful."

Radcliffe's love of literature was given a good grounding even before Harry Potter. He made his screen debut in a BBC version of Charles Dickens's David Copperfield, after his mother put him forward for an audition to improve his confidence. He followed this by appearing in a film of John le Carre's The Tailor Of Panama. With the bulk of his screen work to date adaptations, he's particularly passionate about Mikhail Bulgakov, the Russian author behind The Master and Margarita, having geeked out in his apartment in Moscow and starred in A Young Doctor's Notebook, a television take on several Bulgakov short stories.

There's something old-fashioned about Radcliffe. Unlike many of his generation, he's not obsessed with social media. He's not on Facebook and he doesn't tweet. "Apparently there are a few people who pretend to be me on Twitter," he says. "I've been shown one of their sites. They can't f***ing spell, which is what annoys me more than the fact they are impersonating me." His main problem with the micro-blogging site is just what it represents. "The world has an attitude now - if you can't make your point in 140 characters, you haven't got a point worth making … I think that stops proper discussion."

It would be unfair, however, to paint Radcliffe as out of step. He loves music, both listening to and playing it; his Harry Potter co-star Gary Oldman having taught him to play the bass guitar.

Every time I've met Radcliffe, he's told me about a new album he's just bought or a concert he's just been to. This time the conversation turns to Metallica, and the recent film Through The Never that stars the band and Radcliffe's Kill Your Darlings co-star, Dane DeHaan. "I do like Metallica. [Regarding the film] I did say to my mum and dad: 'Probably not for you.' They love Dane, but they're not Metallica fans."

When Kill Your Darlings premiered at the Sundance Film Festival earlier this year, the immediate furore surrounded Radcliffe's kiss with DeHaan, who plays Lucien Carr, the enigmatic fellow student who introduces Ginsberg to the other budding Beats. It seems like just a few years ago that he was enjoying his first on-screen kiss - with his Scottish Harry Potter co-star Katie Leung. Any nerves? He shakes his head. "By that point in my career, I'd simulated sex on the back of a horse on stage [in Equus]. So a gay sex scene held no fears for me. And it was Dane, so it was all right."

He recalls the moment of the snatched snog. "We were shooting it at four o'clock in the morning on one of our last days, and we had no time. We were rushing. So it was 'Action!' and me and Dane kissed, and we did the take again without cutting the camera, and then they cut. Five minutes went by and I said to Dane, 'We just kissed - twice!' There wasn't time to be worried or nervous about it." And how did he feel about the end result? "I was pleased. I thought it looked good. It was quite a good kiss!"

He's less forthcoming on his real love-life - something he deliberately keeps low-key. Having once admitted that he lost his virginity at 16 to a much older woman, he's avoided stepping out with anyone super-famous. Linked with both Irish actor Laura O'Toole, whom he met on Equus, and Rosie Coker, a set painter on the final two Harry Potter films, he's most recently been associated with Erin Darke, another Kill Your Darlings co-star, who - in keeping with tradition - is some five years older than him.

It can hardly be easy, trying to date when your every move is watched; hours after we meet, at the after-party for Kill Your Darlings, Radcliffe is rushed into a tiny room with his director John Krokidas and a select few. Krokidas emerges to sing karaoke at one point - a rousing rendition of The Killers' All These Things That I've Done. "It's terminal, my love of karaoke," Radcliffe admits. "Eminem is my standard - either The Real Slim Shady or Won't Back Down." He stays hidden, though, claiming he didn't expect the party to be quite so throbbing. "If I got up there, it would be a sea of camera-phones. And that's not fun."

You can understand his reticence: Radcliffe impersonating the Detroit rapper would be a YouTube sensation in nano-seconds. But he's managed, so far at least, to live his life without being a recluse. During his West End run in The Cripple Of Inishmaan, he says "the crowd was fine", referring to those waiting patiently outside the theatre door every night, but the paparazzi proved annoying. "At least two or three nights a week, they'd follow us home. I'd be like, 'I'm going home! I'm going to bed! I've done a show. It's Wednesday. I'm not going anywhere. Where do you think I'll be going?' That got a little bit tiresome."

It didn't help that he'd emerge every night looking worse for wear. "I know I look a mess when I come off stage. Of course I look a mess, and I'm not going to spend 10 minutes trying to iron myself out. I get blood on my face at the end of the play, and it stains a little bit - but I'm not going to spend extra time removing it. I want to go home! So it does annoy you when you've just done a show and somebody [in a newspaper report] says that I'm sparking concern and many people are worried about my pale appearance. No, I'm just pale - this is what my face looks like!"

He sighs, well aware that erroneous stories are par for the course when you are as famous as he is. Back in late 2009, one tabloid ran a story that Radcliffe had been snapped on a mobile phone camera smoking marijuana at a party (accompanied by the headline Harry Pot-Head). "I'm constantly on drugs or dying - whatever they care to write," he says. "That's another thing you have to find funny. You can't give it too much power." Doubtless the amount of narcotics Ginsberg experiments with in Kill Your Darlings will only further fuel such stories.

Another scene in the film sees him dance with a broomstick - a lament for his days on Harry Potter perhaps? "No," he says, wearily. "I see broomsticks all the time - there's a lot of them in the world! So every time I see one, I don't associate it with Potter." He points out that in one of his upcoming films, the supernatural romance Horns, he wears a red-and-yellow zip-up top. "Everyone said, 'It's Gryffindor colours!' There's always going to be something where I'm harking back to [Potter]."

Radcliffe also seems relaxed with the fact that he'll always, even into his twilight years, be associated with JK Rowling's creation. He admits that while there's "pent-up desire" among he, Grint and their co-star Emma Watson "to play a lot of other parts", none of them wants to shake off the image that defined their teens. "It's an amazing thing to be associated with and to have the support of that fanbase is extraordinary, but I definitely think we all want to be seen as actors rather than characters."

While there are various Harry Potter spin-off movies mooted - including a possible adaptation of Rowling's book Quidditch Through The Ages - Radcliffe shrugs. "I don't try to second-guess what Jo Rowling's going to do," he says. Indeed, it would be foolish to revisit Harry so soon when his career is flourishing - although in Horns, where he plays a young man who discovers two protrusions sprouting from his head, it's not so far removed from Potter world. "It would seem he has a little devil in him after all," noted trade paper Variety, upon reviewing the film.

He will also stay in the fantasy zone for a new version of Frankenstein, co-starring with James McAvoy, who plays the creator to Radcliffe's assistant Igor. "It's such a mad script. It's madder than you'd expect coming out of a studio," he says, with typical boyish enthusiasm. Beyond that, he's just finished The F Word, his first romantic comedy, co-starring Zoe Kazan and Rafe Spall. "It was nice to do a film where at no point did I have to get covered in blood or break down or have a massive fight."

Then there's a reunion on the cards with his Woman In Black director James Watkins for Gold, a film about 1500m runner Sebastian Coe (which, given how fast he can sprint to the toilet, should be a doddle).

"The end goal is to be considered an actor rather than a character," he reiterates. "I think I'm quite a way towards that. I can only judge by the scripts I've been sent. But so far it seems to be the case that people aren't pigeon-holing me."

From a boy wizard to a Beat legend, from Frankenstein's assistant to an Olympic athlete - that's about as varied as it gets. n

Kill Your Darlings (12a) opens on Friday.