I have brought Alexander Skarsgård a small jar of pickled herring. It is from Ikea, so not exactly gourmet, but he is gratifyingly appreciative all the same. His face splits into a wide grin as he turns the jar over in his hands. "You went to Ikea?" he says, making me blush like a schoolgirl. "Oh man, thank you. I'm going to have some right now." He unscrews the lid, proffers the jar in my direction and stabs at a piece of fish with his fork. It looks gray and pallid. "Obviously it's better if you pickle them yourself," he says, popping the morsel into his mouth. "I love the purity of the regular stuff, when it's just pickled with herbs and onions. I hate the fruity, sweet varieties."

We are sitting in a spiffy bistro, just off the Bowery in New York's NoHo, and the incongruous presence of a celebrity vampire – Skarsgård's profile in the US rests largely on his role in HBO's lusty drama True Blood – is creating ripples of interest. At 6ft4in and shamelessly handsome, it's hard to ignore him. A young girl interrupts to ask his name. "Alex," he replies, "What's yours?" "Emma," she says, before racing off to confirm to her mother that, yes – it is the man from True Blood. A waiter approaches to congratulate him on the latest episode, before recommending the potato pizza with truffle oil and fontina cheese, a house speciality. We order one between us. There has to be wine, too, though Skarsgård agonises momentarily. "I got here two weeks ago, and I haven't been sober one day since," he says. "It's not like I'm wasted, but every single night there's been something. In LA you have to plan, like, 'All right, next Saturday, let's get drunk and let's not drive – we'll arrange a car.' In Stockholm or New York you go out, you have a late lunch, you end up ordering a bottle of wine, and someone shows up, you order another. I love that, just the flow of it."

There's a fresh, unguarded quality to Skarsgård. He's not yet so wary of journalists (or too primed by publicists) to have lost his spontaneity. Although he lives a lot of the time in LA, he gets back to Stockholm as often as possible, as if to keep his ego in check. Fellow Swede Jonas Åkerlund, who cast Skarsgård as Lady Gaga's paramour in his 2009 video for "Paparazzi" – he pushes her off a ledge, she returns in a wheelchair to poison him – describes it as a "country cousin" mentality, which turns out to be a compliment. It means that Skarsgård is incapable of affectation, and it explains why he doesn't get mocked or disparaged back home. "Swedes tend to judge very easily; nothing really impresses them," says Åkerlund. "It's really hard to find the right balance, and the only way to do it is to be the guy you are, and never forget where you come from." Skarsgård is not the first in his family to have managed that balancing act. In that respect, at least, he is just like his father – "one of the coolest guys in Sweden," as Åkerlund calls him.

The cool guy, of course, would be actor Stellan Skarsgård, who seems to glide effortlessly between theatre, art-house movies and camp blockbusters such as Mamma Mia! But his position as the most famous Skarsgård on the planet is looking shaky right now – at least in the US, where True Blood has established itself as HBO's biggest hit since The Sopranos. Young Skarsgård says it was only in the past year or two, during the second season of the series, that he began to realise his career was taking off, some eight years after his dad's manager suggested he audition for Zoolander, but he bats away the suggestion that there might be any oedipal rivalry in progress. "We're more like brothers than father and son. We hang out. I'll take him out with my buddies in LA or in Stockholm, and it's never awkward or anything. He's 60, but he likes to party." You get the impression that this is something that comes easily to both of them.

“My plan was never to be an actor…”: the Skarsgård clan, from left, Alex, Stellan and Bill. Photograph: Jeff Vespa/WireImage

Dad will soon be coming to visit his son in New York, where Skarsgård is filming What Maisie Knew – an adaptation of the Henry James novel in which he stars with Julianne Moore and Steve Coogan. It's the latest in a series of diverse projects that is raising his profile in Hollywood and captivating gossip hacks and bloggers ("Alexander Skarsgård takes a walk in New York City" – Socialite Life; "Alexander Skarsgård Goes to the Gym in NYC" – Just Jared). A remake of Straw Dogs, in which Skarsgård has the dubious distinction of reprising the infamous rape scene with his now ex-girlfriend, Kate Bosworth, opens in the UK in November. By that point Skarsgård will be back in the studio to film season five of True Blood, in which his role has rapidly expanded to accommodate his exploding popularity. It's clear that he is going places, but he's canny enough to know that where he goes depends on the choices he makes. Lazy Hollywood casting agents, it seems, already have him pegged. "Everyone wants me to play Eric Northman from True Blood in a movie with a different name, basically – strong, tough, alpha-male parts," he says. "That's fun to do, but you want to balance that out."

Enter Lars von Trier, the kind of director who can be relied on to round out an actor's resumé. In an act of mischievousness, he has cast both Skarsgårds as disparate buddies in his new movie, Melancholia – a typically dyspeptic outing for the Danish director that culminates, fittingly, with the end of the world. They play the groom and his best man at the wedding from hell. Skarsgård senior, a von Trier veteran who starred in Breaking the Waves, Dancer in the Dark and Dogville, is brash, arrogant and malicious; Skarsgård junior is genial, tender, and so utterly guileless that you ache to save him from the insult and ignominy he suffers. (In an email, Kirsten Dunst pithily identified the qualities he brings to the movie as "grace, humour and love".)

Von Trier has a reputation for putting his actors through the wringer. John C Reilly walked off set during the making of Manderlay, supposedly over the director's insistence on slaughtering a donkey (the scene was subsequently cut), and Björk's bitter fall-out with von Trier after making Dancer in the Dark – she said that he hated women – hardly needs a recap. Skarsgård seems to have suffered no such humiliation. He uses the words "paradise", "heaven" and "beautiful" to describe the experience. Von Trier himself – last seen in Cannes expressing an affinity with Hitler – is "a very sweet man" who "lets you discover your character and the relationships and the scenes on your own. He doesn't even block the scenes, he just throws the camera on his shoulder and goes, 'All right, let's just shoot, see what happens.' And then, you know, 98% of the first take might be a disaster, but there might be a moment in which two actors meet, where there's a look, an exchange where something happens that you can't recreate, and he'll capture that."

Yes, but what about the 98% that ends up on the cutting room floor? Only a masochist, surely, would put himself through that. It's only later, when Skarsgård recounts a gruelling story of his service in the Swedish military, that it makes sense. He was 18 at the time, and had to spend 10 days in the woods – in Sweden, in January – in order to get a hat (yes, you read that correctly). "It's like a fisherman's hat with a brim around it," Skarsgård explains. "And you've been dreaming about that hat, because for the first six months you don't get to wear it, and you see the guys that have been there for a year already, walking around with their hats, and you're like: They all look like Clint Eastwood, they're the shit!' You've got this stupid little baseball cap, and you're like, 'One day I'll have that hat.' And then you do the 10-day test and your feet are bleeding, you are crying because you are so exhausted, and finally we get back to the base, and we think this is the moment we're going to get our hats, and the sergeant is, like, 'All right, that's a good start. Now you're going to go out on a 15-mile run with your backpack and guns and everything.' Grown men start crying. We were almost hallucinating because we were so tired. You look at this 15-mile run, and you think, 'I can't do this, I've got nothing, the tank is empty.'" The upshot, of course, is that he gets the hat. The final 15 miles was just a ruse – the hats were waiting for the men only a mile away – but Skarsgård continues to draw on the lesson he learned that day. "People were so exhausted, but when they saw those hats they ran back to base. That's the moment when I realised, 'When you are out of energy, there's more in the tank.'"

“Everyone wants me to play strong, tough, alpha-male parts”: as Eric Northman in True Blood. Photograph: HBO/Everett /Rex Features

In movie terms, a Lars von Trier film is a little like that hat, a prize of sorts that challenges actors to draw more deeply from the tank. For Åkerlund, the creative hunger evident in both Skarsgårds is what sets them apart from many of their peers. "It's so easy to get caught up in the American system, where everybody tells you what to do, and what not to do, and where the rest of the world is not important. A true artist, in my mind, is willing to fail sometimes, because if you're not brave enough to say yes and follow your gut, it's never going to be good." Skarsgård considers the two months spent on Melancholia as a revelation. "I think we all felt that this was why we all wanted to be actors," he says.

This is a turn-about for someone who long believed that acting was precisely what he wasn't going to do. Skarsgård more or less grew up backstage, watching his father at Stockholm's Royal Theatre, but a stint as a child actor was instructive. "I can deal with it now, but 13 is a tough age to be recognised and famous," he says. "It's a tough age, period. I wanted to spend 100% of my energy figuring out who I was, and what was happening to me, and it freaked me out to be talked about in magazines or on television – this is who he is, this is what he likes. It made me feel insecure and nervous."

He turned to the one person qualified to give him advice: his father. "He just said, 'I love my job, but it's a tough job, and 99% of actors can't support themselves financially, so it has to be worth it; you have to feel that you have to do it.' And obviously I didn't feel that way. My plan was never to be an actor like my father, so it wasn't a big deal for me to go, 'Fuck this, I don't want to do this any more.'"

It was only many years later, when he was 20, and studying at the Metropolitan University in Leeds for six months ("I loved it"), that he realised he missed acting. "I didn't want to dismiss it, and then, when I was 50, be, like, 'Oh, man, I should have tried'." He enrolled in drama school in New York and then scored the part in Zoolander. The next three years were spent crisscrossing the Atlantic for auditions, before another HBO series, Generation Kill, set during the Iraq war, gave his career wings. Getting to kiss Lady Gaga just as her career was taking off didn't hurt either. "She wasn't that big at the time," he says. "I kind of had to look her up." Skarsgård wasn't so big himself. "I don't think anyone really knew who he was", says Åkerlund. "Of course the record label and management had different suggestions."

Skarsgård says he has to find something of himself in his characters, no matter how disparate the roles. "In movies we tend make things black and white: you're either this, or you're that. Eric Northman is very different from Michael in Melancholia, but I think I have both Michael and Eric inside me, and I think that's what's interesting about human beings – that we're capable of so much, good and bad, and we're fighting that constantly."

Was there a moment in his life that crystallised that lesson for him, that not everything is good or bad? "Yeah, when I saw weakness in my father, I think. When I saw that he was human and that he could be wrong and make mistakes, because I idolised him when I was kid, and he was fucking superman. He couldn't do anything wrong. But you can never connect on a deeper level if you idolise someone – you don't see the real person."

Skarsgård's parents divorced four years ago, but it seems not to have been acrimonious. His father has since remarried and had another child. "They all get along. They have dinner parties every night in his apartment, and friends and family will come over, and Mum comes over as well. Of course, when it happened it was emotional for both of them, but I told them they'd be happier. It was so obvious to me that there was a lot of love there, but they weren't supposed to be together any more; they weren't good for each other."

Although he is largely based in LA, it is clear that Skarsgård's heart belongs to Sweden, where he grew up in the now-popular neighbourhood of SoFo (south of Folkungagatan). His family owns a house on one of the islands in the Stockholm archipelago, and the actor makes a point of going back as often as he can. "There's something I love about how stark the contrast is between January and June in Sweden," he says. "In a way, I feel that time doesn't exist in LA. Sometimes I don't know if it's February or April or October, because you're always sitting outside on the same patio, and it's 70 degrees, and the sun is shining." Of course, a lot of people love LA for precisely that reason, but Skarsgård, who won his hat by surviving in the woods in a Swedish winter, does not settle for easy. "There has to be a challenge there," he says. "There has to be a meaty character. I can't just be, 'and then the hot guy walks into the room.' It's not interesting if someone is always perfectly tanned, the hair is blow-dried. The only thing that's important is to make it real."

Melancholia is released on 30 September, Straw Dogs on 4 November