Welcome aboard, says the new Doctor, shutting the door on the outside world. It's a cold, gusty day at Doctor Who HQ in Cardiff. I'd been hoping for the Tardis, but we make do with his trailer.

The heater is blowing out hot air, and photos and postcards are flapping on the wall. There's a picture of a cat, a mate doing a moony, good luck messages. The trailer has become home for Matt Smith while he's been filming his first Doctor Who series.

He already feels he's done his share of time travelling. Friends, strangers, future fans ask him what it's going to be like when he's the Doctor, and to answer he has to do a double take – take a trip back to the future. As far as he's concerned he's been busy Doctoring for the last seven months, but of course we're yet to see the fruits of his labour.

The announcement that Smith was to take over from David Tennant as the Doctor was greeted by the nation with a huge collective question mark. Matt Smith, who he? After all, names such as James Nesbitt and David Morrissey were said to be in the running. There had also been hints that the BBC was preparing for the first black Doctor (Paterson Joseph, who played Rodrick in the series) or the first female (former assistant Billie Piper).

It's hardly surprising that the supposed shortlist was high profile. Doctor Who is the biggest time-travel series in the galaxy. It's the longest-running sci-fi show on television, sold to 54 countries, and more than three million Doctor Who DVDs have been sold. Previous incumbents have included long-scarved legend Tom Baker and white-haired dandy Jon Pertwee. To make matters worse for Smith, Tennant had just been voted the greatest ever Doctor.

Matt who?

Today he's wearing braces and a bow tie, which make his baby face even more incongruous. After all, so many Doctors were middle-aged at least (even a little wizened in the case of William Hartnell) – and, so we thought, they should be if they were single-handedly going to save the world from Daleks, Reapers, Cybermen, the Slitheen, the Sycorax, the Abzorbaloff, the Master, the Adipose and the odd werewolf.

Smith looks a good decade off shaving, his complexion is worryingly white and he's beansprout thin. His quiff juts out from his high forehead like an overhanging cliff. He's got a touch of the rock-star-from-outer-space about him. "This is my cat Timmy who's no longer with us," he says, giving a guided tour of his trailer walls. "My mate Timmy. I can't tell you, man, I luuuuurved that cat. He got to 18. He had a really good innings, that's why I was so attached to him. He was called the godfather of the street because he was a bigggggg cat. I'm totally going to have a Mini Egg, if that's all right." He digs into my packet and munches away. "Ah, it's like heaven in your mouth."

Words cascade from him, tumbling over each other in the race from brain to mouth. He's bouncy and playful – the type of hyperactive puppy you find presenting children's TV shows on Saturday mornings. Yet at the same time there's something about him that's older than his years. His vocabulary is a throwback to Enid Blyton's lashings of ginger beer: the Ken Loach film Looking For Eric is a "belter" that "warms the cockles", "gosh" is his favourite expletive, he doesn't swear.

When David Tennant announced he was stepping down, Smith's mum suggested he'd make a good Doctor. Why? "Well, I used to have a really stripy scarf at university and people would say it's very Doctor Who!" Is that all? He ums and aahs, and says there might be something else. "I guess I've always hoped… There's something irreverent or wreckless or mad or brilliant. Don't we all hope that we could be the Doctor, somewhere? Isn't that what makes him great?"

Actually, Smith didn't originally hope he could be the Doctor. He hoped – rather, assumed – he would be a footballer. He grew up in Northampton, went to a good local secondary which had formerly been a grammar school, did as much work as was necessary to see him through his exams, and played football. Every day, every evening, he played. He was good, too. Signed on at youth level with Northampton Town, then Nottingham Forest and Leicester City. His dad had been a decent centre-back, and his grandfather had played as a striker for Notts County. There was only one thing he was ever going to be: a first-class centre-back.

Then, at 16, he got a back injury. For a year his father, who runs a plastics business, drove him from Northampton to Leicester every day for treatment. Eventually, he returned to action. But it wasn't the same – the pace, confidence and hunger had gone. Leicester City released him. "I could have signed for a lower league club but it was a risk, and the last few games I didn't want to play, I dreaded it. I'd lost that desire, the urge, the enjoyment, all the things you need in life. I mean, what's the point in doing something you don't enjoy?"

He makes it sound like an easy decision now, but it wasn't. "I was talking to my dad about it the other day. It's the one time… Yes, I was in a mess. Football was everything. You think it's the one thing you do in your life, your whole focus…" For a nanosecond he sounds almost downbeat. "But it's like anything… it's not the disappointment, it's how you react to it." And how did he react? "I went to do my A-levels and started doing drama. I'm totally going to have another Mini Egg, if that's all right."

His drama teacher, Mr Hardingham, nurtured Smith. When he didn't turn up for rehearsal, he gave him a second chance. "Like great teachers do, they change your life. He got me forms for the National Youth Theatre. And I didn't fill them in." Mr Hardingham persisted. Eventually Smith did fill in the forms, won a place, went on to study drama and creative writing at the University of East Anglia and discovered that acting gave him a similar buzz to football. "There are great disciplines from being a sportsman that you can transfer into being an artist. The preparation, the sacrifice, the constant desire to improve."

Smith looks at me. He notices the sweat pouring from my face. It's boiling in the trailer. "Oh God, I'm so sorry, you should have said." He jumps up, turns the heat down, pops a Mini Egg into his mouth, sits down and looks out of the window. He points to the neighbouring trailer that belongs to the Doctor's new assistant, played by Karen Gillan. "She's very beautiful, mad as a box of cats." He stops. "In a brilliant way… We're both pretty mad, I think, which is why it's cool."

Does he fancy her? "Me, Matt Smith, or the Doctor?" The Doctor? "Well, you'll have to see, won't you?" And Matt Smith? "Noooh!" he protests. "She's my mate, Kaz, I just take the mick out of her every day." It's not very gentlemanly to say you don't fancy her, I say. "No, listen, she's a beautiful woman, you know, but we work together – that would be an error. That would be an error!"

Has he always been popular with girls? "I like to think so." Last year it was reported that he was going out with Brazilian singer Mayana Moura. Are they still together? "No, no, no, no," he says in a rush. Smith is single again. What is he looking for in the ideal woman? "Oh gosh! Daisy Lowe is taken, so that's out of the question. Haha!" Who's she? "She's a pretty lady… Oh, I don't know. Someone lovely with a good heart who enjoys the things I enjoy… who plays the guitar."

He talks about how tough the work is, and the hours they have to put in. "By the end, we'll be filming from 11pm till nine in the morning… then they need to shoot in the mornings because of the light. D'you know what? It is exhausting. We've been shooting for seven and a half months now, and the line-learning is quite immense for the Doctor because he's in pretty much every scene, and he says the majority of stuff because his brain is the coolest and the biggest." After filming, he does a couple of hours a night of revision, learning his lines. Is it worse than school? "No, because you're the Doctor, so the payoff's greater. It's not like triple maths with Mr Humzinger. There's not that much coffee breath."

When he got the job, he had to keep it secret. He'd sit watching Doctor Who with his flatmate, desperate to tell him he was the new Time Lord and having to keep schtum. "It was a complete nightmare." Eventually he told his father. "He was rather flabbergasted. When I told him, he laughed. He was excited, elated and very proud."

It's such a strange time in his life. A year ago, he was pretty much unknown – fans of the television series Party Animals, in which he played a parliamentary researcher, or those who had seen him at the Royal Court in That Face, playing the carer son of Lindsay Duncan's alcoholic mother, might have been able to put a name to the face, but he hardly had a mass following. At Christmas, he made his first (and so far only) appearance as the Doctor, when David Tennant regenerated into him. Today, he can just about get away with walking around unbothered. In a few weeks, he will be a star, one of the most recognisable actors in the country. "There aren't many jobs that change the fabric of your life in the same way – where you go from being a working actor who is pretty anonymous, to being thrust into what is one of the most popular shows, if not the most popular, in Britain."

Is he nervous? Look, he says, it's his job, he's taking it all in his stride. Then he stops. Of course he's nervous. "It's unlike any job I'll ever do because a) there's so much that comes with it, b) there's so much that is expected. I'd be lying if I said the first day I walked on the beach, where we filmed, and I saw the Tardis, and there were all these paparazzi there, and you're going, what the hell is going on…?" In his rush to get the words out, he often forgets to finish sentences.

Has he sought the advice of former Doctors? He tells me he recently had lunch with Peter Davison, who told him simply to enjoy the ride. He also had a word with David Tennant. "I spoke briefly to David. He was just very lovely and gave me encouragement, but I think you have to cleave it out yourself. It's your own journey."

Surely it makes it that bit tougher when he's following the most popular Doctor ever. "Yeah, yeah. I guess you've got to approach it with your own take or spin. No, spin is the wrong word. Identity. How can you not be aware of the rich heritage and legacy? Over Christmas it was everywhere. It was the big thing, David leaving the show. But it only intimidates you as much as you allow it to."

I tell him about my 10-year-old friend Joseph. Joseph cried when Tennant left. More to the point, he cried when he saw Smith for the first time. I feel mean mentioning it, but perhaps it's a measure of Smith's task, to win over the fans. He looks a little crestfallen. Then there are the bloggers, complaining that, at 27, he is too young, too much like an emo rock star, too pretty, too ugly, too everything. "Thanks for reminding me." How can he put them at ease? "What is my retort to them? Thank you for your kind and considered comments… hehehehe. Everybody's entitled to an opinion. But an emo? Maybe it's my hair."

Back to Joseph. He wants to know how Smith is going to make the Doctor his own.

"He's a little reckless. He'll walk into a room and have a million things to do. And, as opposed to knowing exactly how to get out, he'll take it up to the precipice: don't know, don't know, don't know, and boom, there's the idea. And it's a bit mad and reckless. It's very doof, doof, doof. And he's got a companion who I think is the hardest to handle. And she's quite mad. But the Doctor's quite mad as well. So together…"

Is he going to be one of those melancholy doctors like Christopher Eccleston, weighed down by his traumatic history? "I think it's impossible to escape that with the Doctor. He's lost so many people and devastated so much… bad or good, he's brought whole empires down. He's seen a lot, and that's part of his personality. But that's also what gives him such joy and effervescence."

As he talks I notice a long scar running down his forehead. Is it the result of a football injury? "No, I cracked that open as a kid at nursery, and had to have 24 stitches. Can you see where it goes back? You can peel my hair back if you want." So I do, and tell him this is not the first actor I've got intimate with – I once pulled a hair out of Daniel Day-Lewis's ear. He looks impressed. "Really? He's a genius. I'm having another Mini Egg. What an actor! That's what we're all aiming for. To turn in bits and bobs like that!"

Apart from the football blip, you sense Matt Smith is used to achieving the goals he sets himself. At school, he decided he wanted to be head boy, and won the election. "I was the outside choice. I curried favour among friends. I just wanted the mantle." Really? "Yeah, course, man. I wanted to run things. I wanted control. It was the highest position, and I wanted the highest position. Why not? Then you get to organise the ball and you get to say if you have a year book and every Tuesday you go out for a meeting for two hours and get out of triple maths. There was a big hoo-ha with a mate called Dean, because he thought I was using underhand tactics to get votes." But get the votes he did.

Whereas previous head boys were straight-A students who had gone on to Cambridge University, Smith was not a swot. But even when stressing his averageness, he ends up revealing more about his determination. "I got an A, B and C. A in Drama. B in Psychology." He bangs the table. "I should have got an A, man. Should have got an A. That ticked me off!"

At university he went on to get a very average degree. He squirms when he mentions his 2:2, but adds by way of explanation that he spent much of the last six months away from his course acting professionally. Just to think, he says almost wistfully, he had once planned on doing a degree in politics at Sheffield.

Why did he want to study politics? "I just wanted to be intelligent, or seem intelligent, and go to a redbrick university and go, 'Oh yeah, I'm doing politics!' I used to read Gore Vidal books and think I was cool."

What would he rather be, a political thinker with a good degree or the Doctor? He looks at me as if I'm absolutely bonkers. "Come on!" he shouts. "Politics was a ridiculous idea. I would have been rubbish. It's the Doctor!"

• The new series of Doctor Who starts on BBC1 at Easter.