Matt Berninger, the lead singer of The National, is at home in Brooklyn. "It's surprisingly cold and I don't know why. Every time I'm home it's unseasonably cold," he sighs. There's plenty of sighing in The National's songs but, apart from complaints about the weather, Berninger is as ever upbeat, polite, talkative and thoughtful. His band, who have been playing with increasing critical and commercial success over the past decade, are about to release their fifth album High Violet, after streaming it for a few days on the New York Times website.

Of course the New York Times doesn't just stream any old thing, and the fact that The National, formed by friends from Ohio and based in Brooklyn, can command that kind of access from media outlets is testimony to where they're at. Revered and lauded, they possess that aura of 'can do no wrong' like The Boss, or Prince, or Jay Z. In an era of indie disposability with next-big-thing hyped bands coming along more often than 46As and disappearing just as quickly, The National wait quietly in the corner of the room, propped against a wall nonchalantly smoking a cigarette, cradling a tumbler of whiskey and occasionally stepping out of the shadows to mutter something profound.

"I think they're forced to in a way because we did start 10 years ago," Berninger muses when I ask why he thinks they're treated with such reverence both by fans and the music press. "Most of our records got no attention. Alligator [their third album] got a little bit, Boxer [their fourth] got a little more. That's the reason people now look at us and say 'we can't call them a hype band because they've never been hyped'."

He sounds relieved, as well he might. Never before has hype been such a tool of both creation and destruction. Two American bands I've spoken to recently – Surfer Blood and the Drums – barely existed a year ago. The former got 'big' when a New York promoter (Todd P) tweeted about them and now they're touring Europe. The latter got a write-up on the music blog Brooklyn Vegan and are now the support band for Florence + The Machine's Cosmic Love Tour. It's a fast-forwarded existence that you'd imagine The National, who only signed a record deal when they were in their 30s, would be quite uncomfortable with. There's is a country stroll towards acclaim, not the motorway.

"We've never been the 'hot thing' on a magazine, until Paste magazine put us on the cover for Boxer," Berninger continues. "But at the same time, we were also selling out Radio City Music Hall. So we've built a following completely on our own. Even if you don't like us or don't like the music, which is totally fair, you have to say it can't be completely empty hype because we've never had any." He chuckles wryly ahead of detailing the next approach some take to describing them. "Then there's this journalist angle that we're an unappreciated underdog. It's not the way I wanted to develop a career, but now that we're on the other side of it, we're really lucky. We're finally now starting to reap a little fruit over a long, long, long growth period."

He must feel somewhat glad not to have to put up with all of the transient hyperbole that follows 'exciting' bands around in this decade? "I think we all feel unbelievably lucky that even against our wishes it went this way. We've seen so many great bands that have gotten an incredible amount of buzz and hype and then it goes away. And then there's the backlash, which is much more ridiculous than any overhype at the beginning. It's a weird, difficult thing to be a band and to keep people caring about you. The only way anyone can do it is make the records they love. There's no guarantee, but the second you try to forecast any kind of blogosphere excitement or something, you're already going to be far behind and eventually lost. I don't know how anyone could write anything with that in mind. On one level it's hard in a band, the attention is so fleeting , everyone's looking for the new thing they can send to their friends. But if you stick with it and write good music, eventually people will find it. You can build an audience your own way. It might take a long time and it might be frustrating, but it will happen."

It did happen for Berninger and his bandmates, two sets of brothers; Aaron and Bryce Dessner, and Bryan and Scott Devendorf. And it's happened in quite a substantial way. But nevertheless, The National even manage to make fame feel subtle. You'd imagine any notion of celebrity would be greeted with a raised eyebrow and a mutter of polite disdain. But it's hard not to be big when Barack Obama digs your tunes.

"It was weird on a certain level," Berninger says, explaining how he felt when the US president used their song 'Fake Empire' at both the Democratic National Convention, and again in Grant Park, Chicago on election night.

Previously, The National had sold a t-shirt emblazoned with Obama's face and the slogan 'Mr November', a reference to a National song title and the month of the presidential election. They donated all the proceeds to Obama's campaign. "I always had a sense that a political candidate would want not to attach themselves to that song," Berninger muses, referring to 'Fake Empire'. "It's a song about turning away from reality and creating a fantasy world in your head, going off, getting drunk and pretending you're pursing some kind of escape." Not very D:Ream, then.

"When Obama used it, just the music, it worked really well. It was very humbling, to have a song involved in whatever minute way in that campaign and that movement, so when we heard our song playing right after he won, after he spoke, those things were surreal and it was an honour, that's how it felt." He pauses. "Surreal but amazing."

For now, apart from reflecting on surreal but amazing things, The National are getting ready to tour High Violet. The album was the first they recorded in their own studio, which they built behind Aaron's house. Even though they work every day, Berninger admits their recording style is "sort of chaotic and slow and meandering" and "very time-consuming" but right now they're more anchored and content than ever before, having a studio base to call their own.

"I guess we never even had any money to entertain the idea of investing money into anything like this," Berninger says. "The studio cost us just under $100,000. Between five guys we've never had that kind of extra money to put into anything. In the past we just hoped we could book a studio for three weeks for $1,000 a day. We ended up just wasting money, because we'd inevitably go over the time we had. When you have no money you just build up credit card debt, and so on, but this time we got out of that debt cycle. We got out of the red, and it's the smartest thing we've ever done."

Apart from recording, there is of course the writing, and what sets The National apart are Berninger's marvellous lyrics articulated in studio and on stage in a beautiful baritone. He says he writes bits and pieces here and there, and spends most days pacing his apartment, writing loads of songs at once, never individually.

"I always like the songs that have the combination of very sincere emotional moments with also ridiculously ugly and embarrassing jackass moments," he says. "If I can make that work in a song, that's my favourite moment, that space between beauty and ugliness."

Beautifully ugly, or just plain beautiful, on High Violet, they've done it again, and that underdog label fades even more.