"I was on my way back one night from the play I was doing and I'd broken my toe in the middle of the performance the night before, which was completely mortifying.

Someone on the front row had their legs on the stage and I tripped over and stubbed my toe, and my parents had brought all of their friends to the show and I went backstage in the interval … I had to stay on stage for like 20 minutes after I'd snapped it, kind of holding it together …

"And all the women [in the cast], they were all very dramatic and theatrical, and they were going: 'Oh, get Sophie an ambulance, poor little thing. She can't continue.' And someone said: 'Go and get her mother; her mother's in the audience.' I said: 'Don't go and get my mum - she'll do the whole "my baby, my baby" thing and I can't handle that. I just need to be able to get my shoe back on.' Meanwhile my toe is growing to the size of a small country - I'm in hysterics - and someone got my mum. And she came in and I went: 'Oh God,' and she goes: 'Well, Soph, it's Friday night and a full house, better get back on stage. Doctor Theatre will kick in. Have faith …'"

Sophie Kennedy Clark is talking. Sophie Kennedy Clark is always talking. At 23, the actor, sometime model, granddaughter of Calum Kennedy - yes, that Calum Kennedy - and quite possibly something of a star in the making, doesn't do silent.

She chats constantly as she's getting her pictures taken (about, among other things, her mum and dad, her brother and sister, and her dad's fish business in Aberdeen). She gossips away as we walk to the restaurant on this fine, dry Friday afternoon in London, this time about the hazards of wearing corsets. After only the slightest of hesitations, she tells me she regularly plunks herself down on the toilet seat while her mum is in the bath for a catch-up. There is no mute button.

Why should there be? She has a lot to talk about. This is her moment. You may have noticed her when she turned up on our TV screens as David Tennant's eldest daughter Tanya in the BBC drama Single Father. Or perhaps you caught her with Johnny Depp in Tim Burton's vampire movie Dark Shadows. But if not, no matter. She'll be hard to miss in the next few months.

This week you can see her in the cinema in Stephen Frears's movie Philomena, based on real-life events, in which she plays a mother who has her son taken from her by nuns at a Catholic convent (an experience "beyond my understanding", she admits). At Christmas she will appear in the new Lars Von Trier film, Nymphomaniac ("which my parents are thrilled about, as you can imagine"), the story of a woman's sex life.

And then there's the corset movie, Eliza Graves, an adaptation of an Edgar Allan Poe story she can't quite remember the original title of (The System Of Doctor Tarr And Professor Fether, if you must know). "I tend to get the weird parts," she says.

In our time together she talks about all that, of course. But, really, she just talks. In Kennedy Clark's world everything becomes a story. As soon as we sit down she's telling me about her birthday party, about the Toronto Film Festival, about not being much of a party animal, about not going to university, about moving to New York and moving in with a bohemian artist in a reformed factory loft in the Lower East Side where her promised bedroom turned out to be a futon in the corner. "I was ultimately living like her pet girl because it was a bit like having a dog basket." About how she chased her agent. "With the arrogance of youth I bombarded him with phone calls, letters. Verging on the stalkerish, I'd go so far as to say." And about how she auditioned for Single Father, the role that was to prove her big breakthrough and the financial implications of said auditioning. "I had to pay my way up to Glasgow. Four callbacks [further auditions]. I was so out of money I thought: 'If I don't get this I'm up s*** creek.'" And all of this comes from me merely asking her how long she's lived in London (three years is the short answer).

"I tell stories," she says. "That's what I do. I've always told stories." She hears them too. Because she speaks to everyone. "Apparently that's a very Scottish thing to do, and so I end up with all sorts of stories."

What follows is hers.

Once upon a time there was a girl. She lived in the country. It was only half an hour from Aberdeen but it felt far from everywhere. She didn't know much about her grandfather's career then. Calum Kennedy, who died in 2006, was the "King of the Highlands", a Gaelic singer - perhaps the Gaelic singer of the post-war years - who graced Grampian TV for much of the sixties and sometimes sang with Fiona, one of five daughters and Kennedy Clark's mother.

Fiona Kennedy was also a TV presence, appearing on the BBC children's show Record Breakers. "People always say: 'Did she inspire you?'" her daughter says. "It wasn't so much that, I think. It just made me realise that being creative could be a reality instead of a hobby. I knew the path had been trodden before so it could be done again."

The young Kennedy Clark spent her time climbing trees and making dens with her older brother and sister. "I was a feral child," she says. "Always on the go. My mother used to sew bells into my dresses so they would know where I was."

Is this making childhood sound like an idyll? Well, it was, she says, but she wasn't allowed to run totally wild. "It wasn't a flimsy bohemian upbringing."

Is she telling me her mother was strict? "Not strict, but she's got her values and they were known around the house."

Her father, Francis Clark, was the practical one. She's taken that from him. "If you're going to get me a gift I'd better be able to use it. Or it must have pockets. I don't get candles or flowers. I think they're ridiculous." He loved films too and so does she as a result.

Mum, meanwhile, was proactive. "If she's not doing music she has something else on the go. I suppose I take that from her. If you're not getting work, make your own work. I think that's a good mentality. I suppose I take my drive from my mother and my practicality from my father."

She was five or six when she told her mum she wanted to be an actor. "I went through a phase where I wanted to be a writer … but it's not social enough," she says.

Writing and acting make sense, of course. She talks in anecdotes. Everything is a story. Here she is telling me about being cast in Philomena, in which she plays the young version of the title character who will grow up to be Judi Dench (I've edited it a bit because otherwise you'd never get to Fidelma Cook's column).

"It's funny. When I first got the audition through in an email - 'Character: young Judi Dench' - I thought: 'Not going to get it. I don't look a thing like Judi Dench.' I go and meet the casting director and it's fine. It's a nice little playful scene. Irish accent. I got a callback. 'OK, meeting the director Stephen Frears. But I'm not going to get it. It's a young Judi Dench.' I go along and it's the nice little scene again and Stephen Frears goes: 'Was that your interpretation of an Irish accent?' I was like: 'Well, Stephen, big fan of your work. Nice to meet you, mate. OK, bye.' Put it out of my mind.

"I got a third callback and they tell me the scene they want to do is where the character has just lost her child. They'd got a big room below the office for me to run around crying and screaming and hysterical, 10.30 on a Tuesday morning. 'Great, I need that like a hole in the head.' I go along and I'm trying to work myself up as I approach the casting director's office. I knock on the door and when they open it they're like: 'We've had a bit of a problem.' I'm like: 'Och, they've given the role away.' And they were like: 'No, we couldn't get the big room downstairs for you to run around crying and screaming in. However we still want you to do something.' I hear Stephen say: 'Oh, bring her in anyway.' So I go into this tiny, tiny little casting director's office and Stephen goes: 'Why don't you just give birth instead?'

"So five minutes later I'm sitting back in a chair, legs akimbo, no preparation … I've seen One Born Every Minute. I'll give it my best shot. And then I did it and felt so overwhelmed by the whole thing that I got up and left. Afterwards I thought: 'I never want to think about this again.'"

She got the part.

Sophie Kennedy Clark is having her photograph taken. "You look like Kate Moss," the PR says. She certainly looks model material. Others have thought so too, though she's not totally comfortable with the idea. "That has been such a tricky thing." She was spotted at Portobello Market in London. "Someone approached me and said: 'Have you considered modelling?' And I said: 'No. I'm five foot six." Next thing she knows she's been signed up by the Storm agency and is shooting an ad campaign with Mario Testino. She didn't tell her parents about it until the billboard was up in central London.

"It was almost bigger than Single Father," she says. And that's the problem. "Everyone goes 'model-turned-actress' and I know there's a bit of a stigma with that. It's not my journey into this industry. But I realise there is an extension of storytelling within modelling and I think these days actresses are on the cover of magazines, not models. People want a story behind a face, and I'm more than happy to do shoots. But I need to have my work as an actress."

We talk a little about Nymphomaniac. Von Trier is the bad boy of arthouse cinema, the man who makes tasteless jokes about Hitler at press conferences and films that are equal parts frustrating and astonishing (the first 10 minutes of his last movie Melancholia are 10 of the greatest minutes in cinema. The rest of the movie, though? Meh). As the maker of Antichrist, Dogville and The Idiots, he comes with a reputation. And no doubt Nymphomaniac, which has a huge cast of names including Charlotte Gainsbourg, Shia LaBeouf, Uma Thurman and Willem Dafoe, all of whom appear in posters for the film displaying - how can I put this? - their sex faces, will add to it.

Kennedy Clark appears in one promotional image in a rather compromising position in front of Dafoe (at least they've got their clothes on). What did she think when she was sent the script? "I have to say it's intriguing if nothing else. And what Lars does so well is he looks at something that is controversial or a taboo and he goes: 'Why is it a taboo? Let's make a film about it and educate people as to why it is a taboo and whether it should be.' And that is within all of his films.

"And ultimately everyone's going on about the sex in the film, but it's probably one of the only things that humans have done since the beginning of time that isn't actually on camera. It's hidden away. It's easier for us to watch someone shoot someone with a gun than it is to watch people having sex."

She can't talk about what happens in the film but it is very clever, she thinks. And why shouldn't someone make a film about a woman's sexual life? "People make films like Shame [directed by Steve McQueen]. Male sex addiction is widely known. When do you hear about female sex addiction? You don't. It's easier to call a woman a slut than to say: 'You have a sex addiction.'"

All she'll say of her own character is that she's a late seventies feminist who's a friend of the main character "and we go on a few adventures". The experience wasn't traumatic or damaging, she is keen to tell me. I guess Von Trier's reputation of putting actors through the mill precedes him. After making Dancer In The Dark with him, Bjork accused Von Trier of having to destroy women in his films because he envied them; Von Trier said working with Bjork and her people was like "dealing with terrorists".

Yet if anything, Kennedy Clark seems rather beguiled by the Danish director. "He's a genius. He's definitely a weird one but he's getting something right."

"The first time I met him he was incredibly softly spoken, very sweet, very interested; all the things you want when you meet, especially someone you're going to be working with. And there are reasons why actors work with him time and time again." Maybe he pushes you, she says, but that can be liberating.

"I've never worked with a director like him. He is one of a kind. I know some people haven't had particularly pleasurable experiences working with him, but I learned a lot and he gives you a kind of creative licence as an actor."

And that's what she wants. Ask her about ambition and she talks about making more movies because it is her favourite thing.

In the meantime she lives alone in London, paints and writes in her downtime, worries that if she goes home to Scotland she might not come back. "At some point I might pull a Tilda Swinton and move somewhere in the Highlands, just isolated and calm. But at the moment I'm living out my raucous youth in London town."

I leave her to it. The sound of her voice follows me out of the building. n

Philomena (12A) goes on general release on Friday.