A little over a year ago, a 27-year-old hip-hop poet from South London brought her theatrical spoken-word piece to the Traverse and walked away with a Herald Angel award.

Kate Tempest's Brand New Ancients shot a bolt of urban electricity through not only the Greek Myths but the entire Edinburgh Festival. "It was lovely, a very intense experience," Tempest recalls, adding with some understatement. "I think we made our presence felt in Edinburgh."

Tempest returns to the capital on Saturday to launch her new poetry collection, Hold Your Own, with her star in the ascendency. She was recently named as a Next Generation Poet, the youngest on the list, while her debut album, Everybody Down, has been nominated for the Mercury Music Prize alongside work by Damon Albarn, Bombay Bicycle Club and Young Fathers. The winner will be announced later this month.

As you might expect from someone whose influences range from William Blake and Virginia Woolf to Wu-Tang Clan, Tempest is hard to categorise. "For some people, I'm a poet; for others, I'm definitely not," she laughs. Born Kate Calvert, she grew up in south-east London, one of five children. She became a poet "by accident" after performing her raps on buses, in nightclub queues, at parties, poetry slams and spoken word events.

"I was just absolutely obsessed with this stuff I was writing, and showing people there was more going on inside me than they would have imagined," says Tempest.

"People have underestimated me all my life. They still do, because I'm unassuming, because I'm a girl, so I had this desperate urgency. I'd go to a gig and instead of watching the person on stage all I wanted to do was get the microphone off them. That feeling lasted for years. It was just blind desperation."

Slowly, it paid off. Tempest was commissioned to write pieces for the RSC and the BBC, and a play, Wasted, for Paines Plough. When she and her friends took over the Old Vic in 2012 for the launch of her first, self-published, collection, Everything Speaks in its Own Way, Scottish poet and editor Don Paterson was in the audience, and was so impressed he invited her to write for Picador. His mentorship proved crucial to Hold Your Own.

"He's a great poet, the real deal," says Tempest. "One thing Don said to me really early on was to see the possibilities of the page. He said, 'You can't perform a semi-colon, but on the page it's really doing a lot.' It took a while to sink in, but I realised how exciting that world is."

Tempest isn't keen on distinctions between "page poet" and "performance poet" - "I find putting brackets around creativity quite reductive" - but acknowledges a shift in her intentions. Hold Your Own was clearly written to be read. "With my earlier work, everything began as a rhyme to speak with rap music. This collection is different, they've been conceived as poems. With the help and direction of Don, they were written to be together on the page, in a way that I've never really allowed myself to attempt before."

Hold Your Own is informed by the story of Tiresias, the blind prophet who appears in Sophocles' Oedipus the King and Antigone. A symbol of duality - he's a man who spends seven years as a woman; he's blind but has the gift of foresight - Tiresias is used by Tempest to frame a series of very direct explorations of love, sex, sexuality and self-knowledge. "There were lots of things I wanted to write about, and I realised that Tiresias would join them together and allow them to have a shape," she says. "This collection is much more personal than I've ever been. I can't quite put my finger on why, but in the past I've been a bit afraid about articulating certain things. It felt important to make peace with all the selves you are in one lifetime and give everything I have to the work. That's what I'm here to do."

Running concurrent to Tempest's evolution as a poet is her musical breakthrough with Everybody Down, an ambitious conceptual rap album which "isn't autobiographical, but comes from lived experience".

Do poet and musician happily coexist? "They're very different processes," she says. "Music is by nature much more collaborative, and that can be really exciting. They satisfy different parts of my creative side, but they're all coming from the same place and are informed by the same things. It all begins with the same urge."

The Mercury Prize nomination "is a big deal," she says.

"I was trying to battle my way into the music industry for 10 years, then I made this record and the industry said - in much the same way as the poetry establishment has - 'You've proved it, you've paid your dues.' It's extremely satisfying and exciting.

"It means we can play bigger gigs and do more interesting things with our shows and on the next album, all the things I've been desperate to happen for the past 12 years."

Never one to shirk a new challenge, Tempest will revisit the characters featured on Everybody Down in her first novel, planned for publication early in 2016.

"It's a move towards me not having to be there all the time to deliver the work," she says.

"A novel will go out and live without me.

"Who knows, maybe I can have a holiday."

Kate Tempest launches Hold Your Own at the Bongo Club, Edinburgh, on October 25; Everybody Down is out now on Big Dada.