Emilia Clarke is the star of Game of Thrones who wants to appear in Ibsen. Now she's appearing alongside Jude Law in one of the best films of the year. She talks toabout Dom Hemingway, Dothraki rituals, and the pitfalls of Hollywood dating

June 2013: the internet announces the engagement of Emilia Clarke, 26, actor known for her role in HBO fantasy epic Game of Thrones, to James Franco, film star, director, writer, thinker. Text messages fly in from Clarke's friends, some of whom she hasn't spoken to since she was about four years old. "I had my aunt from America calling me up and being like" – Clarke slips into a brassy East Coast accent – "'Where's the ring?'"

She lets out a peal of laughter so gleeful you can almost hear the exclamation marks. She had met Franco only twice when the gossip sites, intoxicated by the sight of them visiting an art fair together in New York, announced their utterly nonexistent betrothal. "It's hilarious," says Clarke. "He is, of course, beautiful. But I feel there's only a handful of women who could form an engagement after two meetings, and I'm not one of them."

This is what happens when you are Hollywood's latest crush. Clarke's Game of Thrones role as Khaleesi, the ivory-haired warrior queen, has transformed her, in an eyeblink, from drama-school graduate to red-carpet beauty and potential Mrs Franco. In the two years since landing the part, Clarke has had an Emmy nomination and a lead role on Broadway; she has travelled the world to shoot Game of Thrones (Croatia, Morocco, Iceland); she has dated one of the wealthiest men in Hollywood (Seth MacFarlane). And now she's in a film with Jude Law, which is as good way as any of saying that life is pretty great.

Today she's in Soho and proving so engaging and confidential it's easy to imagine, as we sip cucumber-infused iced teas, that we've stopped for a quick refresher in the middle of a shopping trip. Brown of hair and a slight 5'2", she's not at all as recognisable as her famously blonde TV persona, and Clarke's easy manner, too, would surprise her Thrones fans. Khaleesi – Daenerys Targaryen, Queen of the Andals and the Rhoynar and the First Men, Mother of Dragons – is not given to guffaws. In three seasons of HBO's dwarves-and-swords epic, I'm not sure we've seen her so much as titter. Which is fair enough when you're commanding an army of freed slaves and rearing three adolescent dragons to burn fiery vengeance on your enemies. It's not a laugh-a-minute job.

If you haven't seen Game of Thrones, a quick word of explanation: Khaleesi is one of the main contenders vying to rule the fictional kingdom of Westeros from the Iron Throne. Her compassion and liberal values (well, liberal for Westeros – she'll still kill you if you stand in her way), not to mention her exceptional cheekbones, have made her one of the most beloved characters in the show. And if the success of the fantasy epic has defied, then eviscerated, all expectations, so has Clarke's own rise. Her father, a theatre sound designer, had attempted to prepare his daughter for the life of an actor. "He told me: 'There's just one line you need to learn: Do you want fries with that?' He was trying his damndest to be realistic, show me that it wasn't Disney."

Emilia Clarke with Jude Law in Dom Hemingway. Photograph: Nick Wall

Clarke's new film isn't very Disney at all. Dom Hemingway is her second big-screen project, after a well-received performance in the British film Spike Island; she plays the daughter of the eponymous Dom, a profane, explosive, safe-cracking anti-hero who turns up on her doorstep after 12 years in the clink. If the film reveals anything about Clarke – besides the fact that she's just as convincing on the streets of east London as she is crossing the Dothraki sea – it is that she knows how to pick a script, because Dom Hemingway is, frankly, tremendous.

Nominally a gangster film, it transcends any and all of the genre's clichés – a Sexy Beast with a smidge less psychopathy and, arguably, a bigger heart. Hard on the heels of Filth, in which James McAvoy threw his pretty-boy reputation out of the window, Jude Law has beaten his own to a bloody pulp, sporting a broken nose, a gold tooth and a post-prison paunch. From the second the film opens, with a tour-de-force monologue to his own cock, he swaggers and rages his way through it like a Shakespearean tragedian who's found himself unaccountably trapped in Leytonstone.

Still, I say to Clarke, it must have been hard to turn up on set and imagine Jude Law – even this fatter, grottier Law – as your dad. "Yeah, he had me really young," she grins. The pair wrote "prison letters" to each other in character to establish the relationship before filming. "It was really nice that someone of Jude's stature and calibre still wants to do that kind of thing." It was nice, too, just to spend some time in London, because while Clarke may not have really been crossing the Red Waste for the past few years, she's certainly been missing home.

You wouldn't bet against Clarke having grown up in the Home Counties. Her accent (Berkshire) and boarding-school education (Oxfordshire), combined with her use of phrases like "Absolutely bonkers!", must seem to the Americans she works with as quaintly British as a cup of Earl Grey. She was determined to be an actor "far too young to know what I was talking about", although an open-day visit put her off the idea of stage school – "all these fierce-looking girls in their ballet uniforms". Clarke's "please-be-my-friend outlook" was, she says, far better suited to the enthusiastic pursuits of dorm life. "There definitely was an element of: 'Golf, why not? Archery, yes please!'" she laughs. She even coxed the boys' eight, until the day she steered them down the wrong tunnel and got them disqualified at a Henley regatta.

Her game-for-anything attitude has served her well: not many actors take their first proper job on an (estimated) $10m TV pilot, with screen training that largely consisted of a Michael Caine video explaining how to hit your mark. When Clarke signed her five-year contract for Game of Thrones her entire acting experience comprised a walk-on part in Doctors and a made-for-TV monster movie called Triassic Attack.

Emilia Clarke: 'She can be a bit of a clown.' Photograph: Patrick Fraser for the Observer

Iain Glen, who plays Ser Jorah – Khaleesi's chief adviser – remembers Clarke's first day on set well. They were shooting a scene on horseback in front of 300 extras when Clarke lost control of her horse: "It went off with her into the high reeds and I think it slightly freaked her out!"

"You ask someone: 'Can you ride?' and as an actor you're trained to say yes," smiles Clarke. "But it's one thing doing some riding on a Welsh holiday with your family… Luckily Iain managed to calm me and the horse down at the same time."

Khaleesi's story – that of an innocent young princess, married forcibly into a brutal tribe of nomads, who grows into a fearsome leader – required Clarke, at 23, to do full nudity and rape scenes, not to mention an initiation ritual in which she ate a horse's heart (the shoot took all day, and she ate 25 of them – "They tasted like congealed jam, with a hint of bleach"). She has also, for four seasons, shot her scenes entirely separately from the rest of the cast and crew – due to the fact she spends most of her time on a different fictional continent. So in London pubs she's regularly bumping into fellow cast members she's never seen before, and only met Charles Dance for the first time last week, at a photoshoot.

"It was a massive role for a young actor to take on," says Glen, "and I know the producers were nervous of the big arc that the character would have to take. Emilia herself would admit she worries about stuff, but it gives her focus and it's not a destructive worry. I just found her a delight, and eager to learn." Both actors admit that the relationship between Khaleesi and her adjutant has had a parallel in Glen and Clarke's own friendship, and Clarke describes Glen as "my rock and my mentor".

"Well," he says, gruffly paternal, "if she ever did need me, she needs me a lot less now."

Earlier this year Clarke made her professional theatre debut, another baptism of fire: she starred as Holly Golightly in Sean Mathias's Broadway production of Breakfast at Tiffany's. Mathias, who had previously directed Anna Friel in the role in London, was intrigued by Clarke when they met. "Like a lot of actors, the people they are on the inside are not necessarily the people they are on the outside. Emilia's a bit of a clown, really. She's very beautiful and very intelligent. She's immaculately professional, but she's very playful. She's very funny, and she hides a lot of delicacy. She has a beautifully painted mask, and underneath that there's a very vulnerable girl."

Like Glen, Mathias encountered Clarke's "fretful" side – "she takes it all very, very seriously, and she's very dedicated. She hadn't done a play since drama school and here she was heading a Broadway show," recalls Mathias. "It's a big weight to carry on young shoulders." The New York critics were not entirely scathing, but neither were they enthusiastic; the play ran to half-empty houses and closed after 38 performances. "It was definitely… optimistic," Clarke says candidly. "A young Brit girl with no theatre experience decided to take on an iconic American role on Broadway. Maybe I should have thought that through?"

Emilia Clarke in Game of Thrones. Photograph: Paul Schiraldi

Still, she seems genuinely buoyant about the experience, and, says Mathias, handled the disappointment with her typical stoicism. "I maintain that Emilia was very, very good, and I hope she will do more plays. She's very well read – she's clever, there's no doubt about that." And Clarke is keen to stretch her theatrical muscles. "American naturalism is what my indulgent actor side loves: a bit of Tennessee Williams, a bit of Clifford Odets, August Wilson – I would just love to tackle some of that. Or Shakespeare. Or Ibsen. There's bloody loads I want to do!" She'd like to work with young playwrights, too – she was certainly swift enough to get on the phone to her agent when she'd read Richard Shepard's brilliant Dom Hemingway script. "Maybe this is shooting for the moon," she smiles, "but I'd like to be able to look back on my career and stand by every choice I made and say: 'Yeah, I did that for the prestige' – or 'Yeah, I did that because I needed a mortgage.' There's so much talent out there, and I want to go out and find it and work with it. That's my naive, optimistic aim."

In Dom Hemingway, there's a scene where she has to sing to a crowded East End bar, and Clarke admits that the time she spent in the recording studio has given her ideas – music was the only other career she considered aside from acting. Sounds like she's preparing to become a Hollywood triple threat, I say. "But what's the third one?" she replies. "No talent lies in my dancing… Maybe it's speaking Dothraki?"

She may self-deprecate, but Hollywood is already calling. The day after our meeting, she flies out to LA for 10 days of meetings about potential movie projects, and if Clarke's rapid rise has one downside it is that it constantly removes her from her family – the younger brother ("He got all the brains; he's going to be prime minister") and parents she patently adores. Perhaps that's why she's so happy today – she's finally getting to spend some time in her "cottage" in Hampstead, and yesterday her crates of belongings arrived from the States.

"I had an enormous cockroach I named Bob who lived with me in New York and I'm really afraid I've packed him," she laughs. "I'm petrified I'm going to open up a crate and there's going to be Bob's babies everywhere." The Mother of Dragons, scared of a few winged insects? Absolutely bonkers.

Dom Hemingway opens on 15 November