Brap. Blud. Bredren. There’s three words you’ve probably not heard recently out the mouth of a comedian. But then again, just as you don’t get Jamaican patois dropped in to stand-up sets very often, when did you last hear a rapper use the word ‘malapropism’? Or rhyme lines such as, ‘I like my women intelligent; independent’? Doc Brown was a rapper before he decided to jack it in and try comedy instead. And just as his hip hop deliberately swerved away from American ghettospeak towards something more British – reflecting his lower middle-class roots in London – Doc Brown’s comedy likes to sit a little bit outside the box too.

His posh translation of hip hop talk, for example – with subtitle cards showing scrubbed-up versions of his comedy raps – went down a storm when he played London’s Comedy Store recently. Using his ‘visual phrasebook’, he talks the audience through a ‘slang 101’, where he explains just what all those ‘bluds’, ‘whack’, ‘ting’ and ‘whagwans’ mean. ‘I want to create something no one can steal,’ explains the 31-year-old. ‘Copying other people’s styles isn’t what I’m about. But finding your own uniqueness is like your fingerprint.’

Doc Brown, or Ben Smith to his mum, grew up in northwest London, and considered himself a geek. ‘That’s where the nickname came from,’ he points out. ‘Like the Doc in Back to the Future, I was this nerd. The rude boys were always friendly with me, but I wasn’t one of those guys that was in with the in-crowd.’ Round about the time Smith discovered De La Soul, the Wu-Tang Clan and Public Enemy, he also got into b-boy battles, where he made up raps and entered open-mic nights to win money. ‘Actually, that’s probably the only thing that’s as scary as stand-up. Freestyle battling needs improvising, clever cussing, witty put-downs; that was pretty nerve-wracking.’

As he got more involved with the London underground rapping scene, Smith became a member of the group Poisonous Poets, and started getting airplay on Radio 1 and hip hop stations. Although he ended up supporting Busta Rhymes and De La Soul on tour, and was invited to MC for producer Mark Ronson, Smith says he still always felt like ‘a bit of an outcast. Don’t get me wrong, I had some incredible experiences with rapping, but I ran out of steam. Sometimes I felt a bit like the Fresh Prince of Bell End. In my heart I’m a geek and that’s never going to change. The world of rap is very hard and po-faced. I’ve always had respect for street culture, but I think people respect you more if you don’t try and become something you’re not.’

So he turned his back on the chin-out, bravado and balls world of hip hop, for something more honest. ‘The kind of comedy I try to aim for is very candid; I wear my heart on my sleeve. It’s impossible to hide in stand-up. If you try, you just get found out.’ Brown says he only seriously started appreciating comedy two years ago after his older sister, the author Zadie Smith, turned him on to some acts that have since become his favourite comics. ‘Zadie is a massive comedy buff,’ he says. ‘I never knew anything about stand-up. Peter Kay, Jimmy Carr; that was it. She showed me people I never knew existed: Daniel Kitson, Edward Aczel and Russell Kane.’

So does the famous-relative syndrome feel like a curse or a blessing to him, as he tries to put his own very unique stamp on the world? ‘I’m massively proud of everything my sister’s achieved. I guess back in the music days, when I was trying to make a name for myself, and I was constantly being asked about her in interviews, I was like, “Here we go again … ” But I’ve matured and realise I’ve got a lot to thank her for.’

And how does he feel before coming to Edinburgh with his show, Unfamous? ‘Stand-up is a way to talk to different kinds of people and get my character across. The best comedy moments come when our guards are down. Even if people don’t like me, the haters are going to come away knowing a lot more about me.’

Doc Brown, Pleasance Courtyard, Pleasance, 0131 556 6550, 7–29 Aug (not 16), 7pm, £9–£9.50 (£7.50–£8). Previews 4–6 Aug, £5.

Zadie Smith, Charlotte Square Gardens, 0845 373 5888, 17 Aug, 3pm, £10 (£8).