Creator and star of the musical phenomenon Hamilton, Lin-Manuel Miranda is young, scrappy, and hungry. (Well, youngish.) A recent afternoon in New York finds him at the Rainbow Room at 30 Rockefeller Plaza, forking up cod in miso broth. Through the wide windows, midtown Manhattan lies spread out beneath him like a lumpy quilt.

How does the uncool son of Puerto Rican immigrants, raised in the nosebleed section of Manhattan, grow up to be the presumed saviour of Broadway? Miranda doesn’t know, but he isn’t complaining. “It just feels like being fucking queen for the day,” he says.

The show’s influence is huge. Hamilton’s portrait was due to be shunted off the $10 bill. Now it’s back on

That expletive-heavy enthusiasm is typical of Miranda, a performer, composer, lyricist and, perhaps, one of the most lauded artist in America today. At 36, he has an Emmy, a Grammy and enough Tony awards to fill any reasonably sized mantelpiece. But he still maintains a kid-in-a-candy-store attitude toward the opportunities his great success has brought him – the cod, the view, the chance to write a song for a Disney film.

A few minutes after lunch he’ll head into rehearsals for the episode of Saturday Night Live that he’s due to host. And if he has a free moment, he’ll try to track down the musician Questlove, who works in the same building, and together sort out the track order of his forthcoming album, The Hamilton Mixtape, in which his musical heroes respond to the musical. It is out on 2 December, the same the day the new Disney movie, Moana, which he co-composed, arrives in UK cinemas.

He wears a comfortable hoodie and a newsboy cap decorated with a pattern of peacock feathers. He seems a little embarrassed by his patchy facial hair. “This is just not shaving for two days,” he says. “I have a very Puerto Rican face. I can’t grow a beard.”

Facebook Twitter Pinterest Hip-hop history man: playing the lead in a performance of Hamilton in New York Photograph: Joan Marcus/AP

It’s been a couple of months since he famously sheared off his goatee, clipped his ponytail, and played his last performance in the title role of Hamilton, which continues its Broadway dominance. The story of one of America’s forgotten founding fathers set to a hip-hop beat, Hamilton won 11 Tony awards and continues to be the most in-demand ticket in New York. Michelle Obama has called it “The best piece of art in any form that I have ever seen in my life.” Oprah and Beyoncé love it, too. Even Donald Trump might have a hard time insulting it, though Miranda has made that a little easier by composing a song based on Trump’s hateful tweets.

Michelle Obama has called it ‘the best piece of art in any form that I have ever seen in my life’

The Broadway production alone is likely to gross $100m per year and looks set to become a billion-dollar industry before long. But Hamilton is a radical success artistically as well as financially. It marks the first time in decades that a Broadway musical has both echoed and influenced popular song, though Miranda says it wasn’t necessarily his aim to revolutionise 42nd Street. “I write musicals because I love them,” he says, “not because I want to take them down and make something new from scratch. I want to do my version of the best of them.”

Nevertheless, Hamilton has revitalised the Great White Way and it may have a similarly galvanising effect on the West End, when it crosses the ocean next November, and where his first musical, In the Heights, is currently running. Producer Cameron Mackintosh recently confirmed that Miranda will step back into Hamilton’s breeches at some point in the London run.

One could forgive a man who has won the admiration of both Jay-Z and Stephen Sondheim, earned a MacArthur genius grant, and netted perhaps $10m in the past year for letting this go to his head. But Miranda is a man who wears his fame and his gifts lightly. Over lunch he is almost pathologically disposed to give credit and praise to others, which may be a result of some very effective media training or a consequence of a true and rather humble big-heartedness that accompanies his fierce ambition. (Certainly, he is genuinely, almost helplessly, nice. Concerned I hadn’t eaten, he pressed a fruit plate on me. Also, water, coffee and a plate of lobster rolls that no one could remember having ordered.)

Facebook Twitter Pinterest On song: with wife Vanessa Nadal. Photograph: Gilbert Carrasquillo/FilmMagic

Miranda doesn’t get caught up in the social whirl, he says, because his two-year-old son, Sebastian, “is going to wake up at 6am no matter what”. He and his wife Vanessa, his one-time high-school classmate and until recently a corporate lawyer, take turns with the morning routine. And even as his face graces glossy magazine covers his focus is on the demands of his craft. “On a day when we’re on the cover of something, I’m thinking really hard about the wart underneath my foot and whether or not that’s going to mess up my choreography,” he says. How he measures success: “The guy [in your local shop] knows what you want when you walk in the door.”

He’s been pitching to Disney the idea of ‘a Latina princess. Their stuff gets into kids’ brains early’

He’s still that kid from 183rd Street who – just as he writes of Alexander Hamilton in the show’s opening number – “Got a lot farther by working a lot harder / By being a lot smarter/ By being a self-starter.”

“He’s actually not cool at all,” says Oskar Eustis, the artistic director of the Public Theater, which first produced Hamilton. “He is earnest and warm and generous and loving and open.”

Miranda grew up in Washington Heights, a Hispanic neighbourhood on the unfashionable end of the island. His parents, both Puerto Rican immigrants, met in a clinical psychology programme at New York University, and built a loving, bilingual family that emphasised hard work and show tunes. That wasn’t the only music Miranda heard growing up. He also listened to his sister’s hip-hop albums, the gangster rap his bus driver preferred, Disney movie numbers. Salsa and merengue followed him down the sidewalk, so did 1980s pop – sounds he evokes so vividly in In the Heights. “My neighbourhood has always been cool,” he says.

He started writing musicals at Hunter, a highly selective state school, and continued writing them at Wesleyan, a leading liberal arts college. This is also where he founded Freestyle Love Supreme, the rap improv group he occasionally performs with. (Anyone doubtful of his MC skills should watch the extemporaneous video he made with Barack Obama: “The Oval Office, oh my God, I can’t believe I’m there / It’s so much more intimidating than if it was square.”)

Facebook Twitter Pinterest First lady: Michelle Obama hosts the cast of Hamilton at the White House. Photograph: Chip Somodevilla/Getty Images

The success of In the Heights, which won four Tony awards in 2008 and a recent Olivier for outstanding achievement in music, allowed him to step away from his career as a supply teacher at his old high school. But that early acclaim – and the subsequent flop of the genial cheerleading musical Bring It On – didn’t prepare Broadway odds-makers for the radical artistry of Hamilton. A hip-hop musical about the co-author of the Federalist Papers may seem like a joke to those who haven’t seen it. In Miranda’s hands it is rousing, brash, viscerally thrilling. He uses hip-hop because to him that is the music of youth, of rebellion.

He stumbled upon the story of Hamilton almost accidentally. After those first Tony wins, he took a vacation to Mexico and grabbed the Ron Chernow biography to peruse poolside. (That Miranda picked up a doorstopper for his beach reading is a bit of a crack in the naive persona he cultivates.) As he read, he discovered how, in Miranda’s lyrics, “a bastard, orphan, son of a whore and a/ Scotsman, dropped in the middle of a forgotten/ spot in the Caribbean” grew up to be the first secretary of the treasury before dying in a duel, shot by his longtime rival, Aaron Burr.

He immediately understood this founding father’s narrative as an immigrant’s story, hearing echoes of his own father’s story (well, except for the duel) and that of so many who arrive in the US.

Miranda went on to create what Eustis describes as “this very unique animal of a piece that feels both profoundly personal and about the entire country. It’s managed to do what Henry V did for an Englishman, to somehow distil the best of what people believe and want to believe about their country.”

Facebook Twitter Pinterest ‘I was in the wrong ocean’: a still from Moana. Photograph: Disney

The sound that Miranda found for Hamilton whizzes together a short lifetime of musical enthusiasms, melding hardcore and operetta, the American songbook and the British invasion. He combines genres and cultures, creating a pastiche style that is both heavily indebted to the artists he loves and utterly his own. As Thomas Kail, the director of In the Heights and Hamilton, says: “He can write in so many different idioms with real fluency.”

Hamilton’s influence is almost impossible to overstate. The portrait of Hamilton was due to be shunted off the $10 bill. Now he is back on. The musical, with its cast of mostly non-white actors, has made an unbeatable case for diverse casting and for the use of contemporary music. Tickets have been sold for top prices and those unable to score one have been entertained by Miranda’s nimbly upbeat Twitter account, his gleeful talk show appearances, and by Ham4Ham, the free sidewalk shows he devised so that the hundreds who failed to get tickets in the daily lottery could still leave happy.

Miranda left Hamilton in July. He is used to it now, but “the first week was about checking the back of my neck for mic tape,” he says. Though the reported $100,000 he earns in weekly royalties allows him some leisure, leisure isn’t his style. (Another Hamilton lyric that applies to his own life: “Why do you write like you’re running out of time?”) In addition to The Hamilton Mixtape and the forthcoming In the Heights movie, he’s in the upcoming Mary Poppins sequel opposite Emily Blunt, and he’s finishing the music for Moana, the story of a Polynesian princess who leaves her island in search of adventure.

He describes composing for Disney, in collaboration with Opetaia Foa’i and Mark Mancina, as a kind of “full circleness” as The Little Mermaid was the movie that first made him want to write musicals. He skived off school to buy the video on the day that it appeared in the shops. “‘Under the Sea’ really got me,” he says. “I couldn’t believe I was listening to calypso music in a Disney movie.” (His son, Sebastian, is named in part for that song’s crooning crab.)

Disney’s Oliver and Company was also important to him as a kid. “One of the first Latino characters I remember experiencing on film was the Chihuahua,” he says. “That Cheech Marin voice. I loved that fucking Chihuahua, because it was what I had.” So that future Latino kids have role models other than yappy dogs, he’s been pitching to Disney the idea of “a Latina Disney princess – and it’s exciting,” he says. “The stuff they make gets in early, it gets into the kids’ brains at a very early stage.”

Moana has prodded him to expand his musical vocabulary once again, abandoning his Caribbean rhythms for Polynesian ones, learning the virtues of the log drum. “I was in the wrong ocean,” he jokes. His goal for Moana, he says, was to “write something that would be at home next to the rest of the Disney things, but that is completely original and as much in our voices as possible.”

Hamilton has changed Broadway. But it doesn’t seem to have changed Miranda much. As his friend and In the Heights collaborator, the Pulitzer prize-winning playwright Quiara Alegría Hudes, says: “He knows what he wants out of life and in some ways has already gotten that, which has to do with the quality of collaboration and community that he wants around him. I don’t think that his lifestyle has changed much at all.”

Of the financial windfall, Miranda downplays it. “I bought myself time to write,” he says. Miranda still lives near his old neighbourhood, he still hangs out with the same high school and college crew, he still takes the subway. “I’ve written too much good shit on the train to not be able to take it,” he says.

But he will admit to one indulgence. The occasional, late night “guilt-free cab ride”. And OK, one more. When the first big pay cheque came in, he did what any unswerving nerd would. “I went crazy in the Nintendo store,” he says.

Disney’s Moana is in cinemas on 2 December