On a hot day in late August, Lin-Manuel Miranda sits in a lecture theatre at Columbia University in uptown Manhattan, fizzing with the kind of energy that only comes, one imagines, from the experience of creating a billion-dollar Broadway show. Two weeks earlier, Miranda took his last bow before leaving the cast of Hamilton, the rap musical he wrote and starred in and for which the word “hit” seems, at this point, inadequate.

Since its opening last year, not only has Hamilton sold $1bn worth of tickets, won a Pulitzer and 11 Tonys and become the most successful Broadway opening of all time, but Miranda has been credited with everything from reinventing musical theatre to revolutionising the way Americans think about their own history. He has rapped with Barack Obama in the Rose Garden, been quoted by Hillary Clinton in her speech accepting the Democratic nomination for president, and seen his beard and ponytail combo become almost iconic. Such is the pitch of his fame that it is hard, today, not to encounter the 36-year-old, who is slight and boyish and unexpectedly clean shaven, and not simply burst into laughter. What does one do after a year like Miranda’s, I ask. He grins widely. “You cut your hair off,” he says.

The success of Hamilton, which tells the story of Alexander Hamilton, one of America’s lesser-known founding fathers, from the American revolution to the drafting of the constitution, is down in large part to a brilliant cast, made up almost entirely of African-American and Hispanic performers, and told exclusively in rap. In that one act, Miranda simultaneously upended expectations of American history, made a point about inclusion, and also found the perfect idiom for the story at hand; after watching Hamilton, you find yourself wondering how the tale of a scrappy, post-revolutionary nation being born could ever have been told in a medium other than hip-hop.

Facebook Twitter Pinterest Miranda – minus the pony-tail. Photograph: Larry Busacca/Getty Images for Tony Awards Productions

It is rare for a Broadway show to cross over and become cool with other genres of musician; at its height, even a hit such as The Producers had a limited appeal. On any given night at Hamilton, however, so many celebrities were present that Miranda and the cast would joke that “the internet came to see the show tonight”. A single example: “Busta Rhymes coming on the same night as Mandy Patinkin and Salman Rushdie. It was the dream dinner party every night.” (Or, almost every night; there was the one at which Dick Cheney sat chuckling away in the front row.)

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On the day I saw the show, mounted police patrolled the street, the queue stretched around the block and ticket holders looked with pity across the way at the poor sods queuing up for Finding Neverland. Compared with the drama outside, the show itself was, almost inevitably, anticlimactic; original, funny, shocking, but still, you know, a Broadway show, rather than the announcement of the second coming.

It was, nonetheless, breathtaking, a combination of dazzling verbosity and polysyllabic brilliance, combined with the deep joy and bathos of rapping about dull subjects such as the Federalist Papers with the fever of a rap battle: “Thomas Jefferson, always hesitant with the President / Reticent – there isn’t a plan he doesn’t jettison / Madison, you’re mad as a hatter, son, take your medicine / Damn, you’re in worse shape than the national debt is in.” And so on. You get the idea.

As a result of the subject matter, Miranda’s first expectation for the show was simply that it would be popular with teachers. After its New York run, he envisaged it living on as a worthy staple of community theatre. “I thought, if they can get over our use of the F-word, this will be a good school group show.”

Facebook Twitter Pinterest Watch a video of scenes from Hamilton, starring Lin-Manuel Miranda

This is only his second Broadway show – the first, In the Heights, has just been extended in London – and Miranda is still reminded daily of the fact that, 10 years ago, he was a supply teacher. In spite of the fact that his estimated royalties from the show total a mind-boggling $105,000 a week, he still flies commercial – “I don’t care about private jets” – and lives a stone’s throw from his old neighbourhood in Upper Manhattan; the reason we meet at Columbia is that it is around the corner from the apartment Miranda shares with his wife Vanessa and their 18-month-old son Sebastian, who was born two weeks before rehearsals for Hamilton started. (It strikes me that, were Miranda a woman, much more would have been made of this timing – the difficulty of having to be on stage every night with an infant at home. As it is, he says, it all worked out fine because, in spite of being “a very sleepless two years, my wife is a superhero. That’s the answer.”)

Clearly, Miranda works hard to keep himself grounded and surviving the experience has, he says, been a question of “staying on top of the emotions”. Nonetheless, so feverish has the acclaim for Hamilton been, I wonder if, as the frenzy around it grew, he ever thought; “God, what have I done?”

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He shout-laughs. “What have I done?! I have to give enormous credit to [the director] Tommy Kail, who’s great at keeping the temperature low inside the theatre, especially when we moved to Broadway and the yell became a roar. That never came into the studio where we were working.” Instead, he says, they carried on with business as usual, telling themselves: “We’re making our thing better and still have 10 things a day to fix.”

One of the current questions surrounding Hamilton, as preparations are made for it to be rolled out, Phantom of the Opera-style, across the world, is how hard it will be for foreign audiences to love a show that rests, in large part, on knowledge of, and interest in, the American origin story. The life of Alexander Hamilton is a great tale, but the show’s real resonance comes from its interrogation of the American national character.

“I think that if we’ve done our job well and we articulate this individual’s life well, the themes inherent in that translate,” says Miranda. “It’s about legacy, about how much do we do with the time we’re given? And then there are themes that wrestle with the American character, but only in that Hamilton’s life is a rough-draft version of the arguments we still have as a country.” These too, he hopes, will travel. He tells the story of what happened when Fiddler on the Roof made its Japanese debut, after which a theatre-goer approached the producers and said: “They like it in America? But it’s so Japanese!”

Facebook Twitter Pinterest Miranda and other cast members of Hamilton at the Public Theater, New York. Photograph: Joan Marcus/AP

Miranda sees the arguments started by the founding fathers as analogous to his fights with his sister. “The arguments a country has are its family arguments. I fight with my sister and it’s a version of the fight we had when she was 16 and I was 10. I think that’s true of both countries and families. We’ll always argue about the size of government in our lives and the role we play in the affairs of other countries.”

That these dry-sounding debates should animate such an electric show is, of course, a reflection of the scale of Miranda’s talent and his success in answering the challenge set up in the opening line, rapped by Aaron Burr, nemesis to Hamilton and foil to his character: “How does a bastard, orphan, son of a whore / And a Scotsman, dropped in the middle of a forgotten spot / In the Caribbean by Providence, impoverished, in squalor / Grow up to be a hero and a scholar?”

Miranda says: “I think I threw the ball very high in the air with the opening line of the show. This big-ass run-on sentence, and we are going to purport to try to answer it. How does he get from here, to here?” In the first version of Hamilton, which was written for the Public Theater, off-Broadway, Miranda combined song and speech in a much more conventional-looking show. It didn’t work, “because the songs – especially our opening number – is really heightened speech, with melody in it. So when we went to regular speech, you could feel the audience pull out.” At some point, Miranda realised he was going to have to dispense with a script and simply “musicalise every second” of the show, a huge undertaking and one that saw him wandering around his neighbourhood muttering to himself every day, as he tried to fit lyrics to music.

Facebook Twitter Pinterest The London cast of In the Heights, Miranda’s first foray on to the Broadway stage. Photograph: Tristram Kenton for the Guardian

He had never wanted to be a solo rap star, he says. “My ego is healthy, but it doesn’t extend in that direction.” Instead, from school age onwards, he had only ever wanted to be involved in the theatre. His parents had moved to New York from Puerto Rico before Miranda was born and landed in Upper Manhattan, where his mother was a psychologist and his father a political adviser to, among others, former New York mayor Ed Koch.

Miranda grew up in a household rich in intellect but not particularly wealthy and he was aware, at college, that sacrifices were being made in order to educate him. As a result, he was highly motivated and started writing In the Heights, a show about his Washington Heights neighbourhood, before he graduated, slogging away at it for the next six years while working as a supply teacher to pay the rent, until, in what he still regards as a miracle, it came to Broadway. Overnight, “I went from being a professional substitute teacher to a Broadway composer.” It was, he says, an even more “drastic life change” than the one brought about by the success of Hamilton.

In the Heights was a hit, winning four Tony awards, but the one disappointment was that it didn’t bring Miranda the interest or admiration of his heroes in the rap community. As an adolescent, did the part of him that loved hip-hop disparage the part that loved Broadway musicals? No, he says. “There are a few people who only like hip-hop music, and a few who only like theatre music, and the rest of us just like good shit. It doesn’t matter what form it comes in. I think we’re all a lot more eclectic than we give ourselves credit for. And I feel lucky to have grown up in the era when we made mix cassettes. I think it affected how I write scores. Because when you’re making a mix tape for your friends, it’s 90 minutes of music, in the order in which you curate it, so you’re creating the ebb and flow. That prepared me to write scores more than anything else. You want to surprise them, tell them how you feel about them, you want to express the wealth of your taste. How is that different from writing a Broadway score?”

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Can he remember any of his mix tapes from that era? “Yes. I still think of the mix tape my friend Antonia made me that had both Sam Cooke’s Bring it On Home to Me and D’yer Mak’er from Led Zeppelin, which were the first two songs on side B. When I grew up, one of my favourite artists was Weird Al, who taught me that genre is just the clothes of the artist. Instrumentation can change, but we’re all dealing with the same 12 tones and a good melody is a good melody.”

For Miranda, one of the most gratifying aspects of Hamilton’s success has, therefore, been the fact that every high-profile rapper in the US has dropped backstage to tell him how much they love the show. “What hip-hop artists pick up on that your average theatre-goer doesn’t is that I’m using different flows for every character, the way a classical composer would use a different theme in, say, Peter and the Wolf. Everyone raps in a way that is consistent with their character and I’m modelling that after my favourite hip-hop artists. Hercules Mulligan is Busta Rhymes in my brain, and Hamilton is from the school of Eminem and Big Pun, where it’s how many syllables can I rhyme within a line, versus [George] Washington, which is much more straightforward and on the down beat. Because that’s who Washington is. Very regimented and a moral authority. To hear what Eminem, André 3000, or Chris Rock caught that I embedded in there – that is really fun.”

Facebook Twitter Pinterest Song and dance man: an extract from Hamilton being performed at the Tony awards in June 2016. Photograph: Theo Wargo/Getty Images

This, the period after his exit from the show, could have been an anticlimax for Miranda but, for now, he says, he is happy finishing up some back-burner projects – the score for the Disney animated movie Moana and the filming of a Mary Poppins sequel opposite Emily Blunt. And while success this large gets you out of lots of things – “I don’t have to write a jingle to pay rent; I don’t have to do a corporate gig. That part of it is very freeing” – at the same time, “it doesn’t get you out of life. I still picked up my dog’s poop this morning and changed three diapers. I’m still ignoring a call from my dad because he wants me to do something I don’t want to do.” Meanwhile, his wife, who Miranda met when they were both at high school and is now a corporate attorney, finally gets some help from him in the evenings.

Miranda has no idea what his next show will be, only that the one fruitful way to find out is to maintain the right priorities. “I work in a world in which only one in five shows returns their investment. Failure is the norm. So you have to walk away from the years that you spent on something, thinking: ‘I did it for these reasons and I got that.’ Financial success cannot be your goal. But if your goal is, ‘I want to tell this story’, or ‘I want to learn from working with these people’, then things have a good chance of working out.”

It is also a question of understanding one’s own motivation, he says. “I think when you’re making something, you’re trying to fall in love with it; to express the best version of that idea. I think, naturally my subconscious tries to create the high I first felt when I was in Pirates of Penzance in the eighth grade.”

This was, he swears, all he was doing with Hamilton, staging it with no expectations, just to “see how the world responds”. Miranda smiles widely. “And the world freaked out.”