Bola Agbaje is remembering the time all her nieces and nephews came to see her first play, Gone Too Far!, a street-smart story of sibling rivalry that won her an Olivier award. "They didn't really get it," says the playwright, hat tipped across her glossy curls, smile playing across her glossy lips. "But they loved just being there. Since then, they're always asking, 'When are you going to write a play for us?' So for my niece's 10th birthday, I wrote a short play that they all performed at home. I had them all playing around with my things. We called the play Auntie Bola's Makeup."

We're sitting in the bar of London's Royal Court theatre, the smell of Agbaje's strawberry lip-gloss just detectable in the air between us. In 2007, at the age of 27, Agbaje emerged from this theatre's young writers programme to instant success with Gone Too Far! Now she's hoping to repeat the triumph with Belong, a work that tackles themes close to her heart: identity and the African diaspora. "I've always had this play in me," she says. "I just didn't know what it was and how I was going to get there."

Belong was commissioned in 2010 by the British-African company Tiata Fahodzi. They'd been impressed by Off the Endz, Agbaje's second Royal Court play, about twentysomethings from a tough estate facing tough choices. Getting the call meant a lot to Agbaje. "The reason I became a playwright was a Tiata Fahodzi play called The Gods Are Not to Blame. I was blown away. It was set in Nigeria, and about traditional Yoruba culture. Since then, I've wanted to do a traditional Nigerian play, to find a way of celebrating my culture."

Agbaje's parents left Nigeria in the early 1980s. She grew up in Hastings and London, though spent two years in Nigeria after turning six. While Belong puts Nigerian culture centre-stage, it's not a straightforward celebration. The central character is Kayode, a Nigerian-born, British-raised MP who returns to Nigeria to escape the furore that follows his electoral defeat. Once there – to the dismay of his wife Rita, who is half-Jamaican, half-Nigerian and wholly happy in London – he is drawn into the country's own political struggles.

"It's almost like a play for my parents," says Agbaje. "They're having conversations now about retiring: do they go back home or stay here? The play was partly about figuring out where they belong. I grew up being told I was black. Then I would go back to Africa and people would tell me, 'No, you're English.' But the wider question for me is the fact that there are two types of Africans now. There are the Africans in the diaspora, who've grown up in London, America, all over the world, who are now going back home. But the people over there don't necessarily consider them as African. So what defines you as a person: is it the colour of your skin, is it where you come from?"

Belong also dwells on tensions within Britain's own black community, Kayode having told black voters, as Rita puts it, "to get off their lazy arses". Agbaje says this reflects her concern about criticism of black and Asian people by black and Asian people.

"I feel like the black community is tougher on the black community. I read an article in the Evening Standard recently by an Asian guy [Nirpal Dhaliwal], going 'Where's the black uproar because that little girl [Thusha Kamaleswaran] got shot? Why are you black people not making noise about this, but you're making noise about the gunman [Mark Duggan] that got shot by the police?' For me, this is the media allowing people to be outrageous, one-sided and judgmental, because they think, 'You can say that to your people and it's fine.'"

Agbaje's plays may have won her acclaim and that Olivier, but they have also attracted criticism. The fact that she chose to set both Gone Too Far! and Off the Endz on council estates, not unlike the one in Peckham where she used to live, led to accusations of stereotyping the black experience. "With Off the Endz, I got a lot of backlash from the middle-class black community," she says, her smile faltering for once. "Gone Too Far! was my first play, so I think people thought, 'Ooh, it's OK.' Then, with Off the Endz, they were like, 'It's on the main stage, it's set on an estate, and she didn't spell "ends" right.' It was really upsetting. It wasn't like I was trying to go, 'OK, I want to stereotype the black community.' I just wanted to tell a story."

Agbaje came late to playwriting. Her first experience of theatre, at school, wasn't positive: "People would come in and do Shakespeare. They did The Tempest and it was just the most horrible thing I'd ever seen. The actors were spitting so much. I was like, 'Is this acting? I don't want to be involved in that.'"

She went on to do a degree in media communications, then got a job as a social housing officer. She only found out about the Royal Court's young writers programme (other alumni include Polly Stenham and Lucy Prebble) after idly Googling "writing courses" one day. She won a place and loved every second. "Being among people who want to do what you do is so inspiring," she says. "Being a writer is such a lonely career. There's a lot of procrastination. I've got a little office outside my house now because otherwise I would just sit and watch Homes Under the Hammer."

She's still doing the day job, though. "I've gone down from five days to three to two," she says. "I think it gives me the drive not to give up on my dream." And drive she has. Next up for Agbaje is a play "about young women and sex and responsibility" for the Liverpool-based company 20 Stories High, and a mooted big-screen adaptation of Gone Too Far! Her ultimate ambitions lie in film: she wants to write an "epic tale of Africa" for the cinema, but so far the transition is proving a struggle. "They don't care that you've got an Olivier," she says, "or that you have plays on at the Royal Court. And I'm reminded constantly that I'm a –" she makes quote signs in the air – "black writer. They don't see beyond black; they feel that the black experience isn't a worldwide experience, or an inclusive experience. In this day and age, it's shocking."

When she was approached, back in 2010, to write a play for the Tricycle theatre's Women, Power and Politics season, Agbaje said she knew little about politics. She was underselling herself: her work has covered everything from urban teenage alienation to asylum-seekers to the black female experience. I point this out to her and the strawberry smile returns. "Someone reminded me recently that life is political," she says. "Everyday experience is political. Everything about our lives is a debate."

• Belong is at the Royal Court, London SW1 (020-7565 5000), 26 April to 26 May. Transfers to the Royal Court's Theatre Local at the Bussey Building, SE15, 31 May to 23 June.