In the six years since Indhu Rubasingham was appointed its artistic director, the Tricycle theatre has achieved three West End transfers, two Olivier awards, a Liberty human rights award and the distinction of mentoring hundreds of London’s teenage refugees through its Mind the Gap programme. Now, almost two years since it closed for a £7m facelift, the building is preparing for its most dramatic flourish. The Tricycle is no more: Rubasingham is relaunching it as the Kiln theatre.

“It’s an opportunity we could only take at this moment,” she says, visibly excited as she gives me a hard-hats-and-fluoro-tabards tour of the refurbished space, which is still five months from being curtain-ready. “It’s been bubbling in my head and I never thought it would happen but it’s the next part of the story for this building. Kiln as a word is associative with Kilburn. Kilns have a relationship with cultures across the world, they are a physical thing, melting pots associated with heat and cooking.”

To put it another way, Rubasingham is hoping to establish the Kiln theatre as the hottest ticket in town. “People who love us aren’t going to not come because we’ve changed the name,” she laughs. “We can’t have been that good in the first place if that’s true, but this is an opportunity to grab people who don’t go to the theatre. It’s for people who think we were a children’s theatre, or that we run yoga classes, or aren’t sure what we do.”

Under her predecessor, Nicolas Kent, the Tricycle became known for political dramas such as The Great Game and The Bomb, and verbatim, tribunal plays, staging inquiries based on spoken evidence. “I do believe in political theatre,” says Rubasingham. “But I think I express it differently: my politics are that it’s about the story first and the stories we don’t hear, from the people who don’t have a platform. I’m committed to showing the world through different lenses.”

To that end, her new season features six new plays including five world premieres, and one British premiere. The theatre reopens in September with Holy Sh!t by Alexis Zegerman, telling the story of two London couples and their four-year-old daughters, and will be followed by the theatre’s most ambitious project yet: the first stage adaptation of Zadie Smith’s White Teeth, a story rooted in the Kiln theatre’s north-west London neighbourhood. Both productions will be directed by Rubasingham.

“New writing is risky,” says Rubasingham, of her tendency to avoid staging the classics, “but if you really want to be in on the zeitgeist and capture what people are thinking or talking about now, it’s the best way to do it.” She cackles when I mention that her gift for staging “caustic, foul-mouthed plays” has been praised in the Observer. “I love language whatever form that takes, I relish it.”

Next January, a co-production with Tamasha and Live theatre companies will see taxi driver-turned-playwright Ishy Din follow-up his smash-hit 2012 play Snookered with Approaching Empty, a play set in a Middlesborough minicab office following the death of Margaret Thatcher. The Son, the final part of Florian Zeller’s acclaimed family trilogy (which includes The Father and The Mother), will carry the season into spring and will be followed by more new commissions from Samuel Adamson and Anupama Chandrasekhar.

Rubasingham has had much to prove on her “longer, more meandering” journey to become one of the first women of colour to run a major London theatre. “Everything is defined by your race and gender when you’re not ‘normal’ in that picture,” she says, with slow consideration. When I first started, I was often asked, ‘How does being Asian influence you as a director?’ and I’d say, ‘I have no idea!’. Or, I would do plays with so many different writers and then be told, ‘Oh, it’s a shame you’ve been pigeonholed’. I wasn’t pigeonholed – I wanted to tell all these new stories! It’s such a tightrope, you have to be comfortable in your skin and have confidence in yourself. You don’t want to be a tickbox exercise or be accused of being a tokenistic add-on.”

She was born in Sheffield, in a traditional Sri Lankan Tamil household, and studied drama at Hull University after a transformative stint of work experience at Nottingham Playhouse when she was 16. I ask, following the Royal Shakespeare Company’s recent suggestion that a theatre review in the Daily Mail had a “blatantly racist attitude”, whether critics have become less prejudiced during her career. She offers a wry smile and points me towards coverage of her Romeo and Juliet, which she set in Ottoman-era Istanbul at Chichester Festival theatre in 2002, or when she cast David Harewood in The Misanthrope.

“But that journey of frustration and obstacles has informed me and pushed me,” she points out. “It made me better and made me clear about my vision. It’s more difficult if you do not see people like you in the industry, people like you on stage, people like you in the stories that are told. But it’s also the fuel that drives and makes me curious about people’s journeys and cultures, to fight for the unheard and tell stories that demand empathy.”

As a consequence perhaps of running a building and endlessly applying for funding grants, Rubasingham has a tendency to speak in form-filling arts jargon, but her passion and talent burns bright; her current production of The Great Wave at the National Theatre, about the scandal of North Korea’s abduction of young Japanese citizens, has gripped audiences and sold out its run. She has a startling track record for kicking down the door of the established canon with new, original work. “Theatre needs different voices to make it relevant for now and the future,” she says, matter of fact. “It’s exhilarating.”

• This article was amended on 13 April 2018. An earlier version described Indhu Rubasingham as “the country’s first woman of colour to run a theatre”.