It’s been a red-letter year for Zoë Kravitz – and it’s about time. Already in 2015, she’s appeared in YA sci-fi sequel Insurgent, drone strike drama Good Kill and George Miller’s bonkers action movie Mad Max: Fury Road. “I was working a lot for a few years and things just weren’t coming out,” Kravitz explains. “Mad Max, we shot so long ago [in 2012]. My friends said, ‘We know you’re an actress…you’re always gone, and working, but where are the films?’ Even I had a moment of, ‘Where is all this stuff I’ve been doing?’”

Thankfully, the 26 year-old is finally seeing the fruits of her labours. Her latest film is Dope, a hip-hop comedy that sparked a six-studio bidding war when it premiered at the Sundance Film Festival earlier this year. Street-smart, funny and refreshingly honest, it’s about a geeky high-school kid from Los Angeles named Malcolm (newcomer Shameik Moore), who gets tangled up with some dope dealers when all he really wants to do is study and go to Harvard.

“People have said it’s about drugs and it’s glorifying it but, no, it’s just not sugar-coating it,” argues Kravitz, who plays Naika, a girl equally mixed up with the wrong crowd. “It’s about young people, but it’s not made for young people. And I love that Rick [Famuyiwa, the writer-director] didn’t manipulate his story to spoon-feed it to twelve-year-olds. He said, ‘This is the story. Take it or leave it’.”

Comparing the film to Ice Cube comedy Friday, the geek-teen tale Superbad and the classic Tom Cruise vehicle Risky Business, Kravitz was immediately sucked in. “When I read this script, I understood these people, I understood the humour, I understood the world that they were living in. I got it. And it genuinely made me feel something.” The trouble is, she says, films like Dope come along all too rarely. “There’s just a certain kind of black film that…I don’t see myself when I watch these films.”

Usually, says Kravitz, she gets asked to play very stereotypical characters. “We’re always asked to play someone’s sassy black friend. Or in black films, I’m supposed to speak a certain way. Why doesn’t my character have an education? Why does she talk like that? I don’t understand. Or it’s the girl growing up in the ghetto with the child when she’s sixteen trying to get off crack. Or it’s an interracial story, but it’s about an interracial couple; it’s not just a love story.” She sighs. “They’re very specific views.”

While she’s quite right to complain about the paucity of roles on offer, Kravitz has an enviable track record. In just her second film, 2007’s The Brave One, saw her working with director Neil Jordan and Jodie Foster, while studios seem to warm to her edgy appeal. Today, dressed in a revealing white mini-dress and heels, with a gold nose ring, her hair in dreads and tattoos of crescent moons and stars all over her fingers, Kravitz doesn’t seem like your usual plastic Hollywood starlet.

As her surname hints, she has a showbiz heritage. Her father is musician and occasional actor Lenny Kravitz; her mother his model/actress Lisa Bonet, who rose to fame on The Cosby Show (For the record, Kravitz claims her mother is “disgusted and concerned” about the on-going sexual abuse allegations surrounding former co-star Bill Cosby). Her parents divorced when she was 5, and Kravitz, who was born in Venice Beach, California, was raised in Florida and later Manhattan.

“I think both of my parents are unique in the way they don’t live their lives as celebrities,” she says. “They’re both artists, first and foremost. My Mom lives a very private life. So does my father. You don’t really see them in the tabloids or anything like that. I think that’s definitely a decision you can make.” Understandably, she takes any comparisons as favourable. “Both my parents are very strong people. And it’s a compliment to be compared to people who are talented and lovely.”

Raised “a total drama geek”, Kravitz won her first film role when she was 16, in the culinary comedy No Reservations, with Catherine Zeta-Jones. So did the family name get her foot in the door? “It was very easy for me to get an agent and the first couple of auditions,” she admits. “But it’s comforting because the more you work and the better jobs you get, you realise you’re not getting these roles because of your last name. George Miller did not need Lenny Kravitz’s daughter in Mad Max!”

Indeed, there’s little that Kravitz needs by way of a parental helping hand now. Over the summer, she’s been in Atlanta, to film Allegiant – the final two-part film that will follow Insurgent and its predecessor Divergent – and there are two other indie movies on the way: Vincent-N-Roxxy, “a dark love story” with Emile Hirsch that she compares to True Romance, and Viena and the Fantomes, in which she plays the manager of an Eighties punk band.

Kravitz fronts her own band too – Lolawolf; the three-piece hip-hop/R&B outfit have already opened for Miley Cyrus and Lily Allen. With all this success, it’s why she doesn’t really have a Plan B. “When you love something, it’s hard to imagine doing something else. I think I would actually have to be put in a position where someone put a gun to my head [to come up with something].” So is there nothing else she could do? “I sometimes say I would be a farmer. Seriously!”

Dope opens on September 4