There's a moment in the current production of Much Ado About Nothing that I particularly love. It's not when David Tennant's Benedick makes his entrance as a sun-bronzed prat in a golf buggy, nor his Cary Grant-style rat-a-tat-tat delivery of lines such as, "I would my horse had the speed of your tongue", not even when he finally gets to kiss Catherine Tate's Beatrice. It's the encore when Tennant skips across the stage, grinning from ear to ear, and you just know there's nothing on earth he'd rather be doing. I've never seen an actor so happy in his job.

Today, Tennant, 40, looks like a teenager posing as an adult. Here he is in his skinny jeans and Beatles T-shirt, bubbling with enthusiasm, and absurdly boyish. He's obsessed with hacking, can't stop reading about it. The police, the press, the politicians all in it together – perfect conspiracy. "Rupert's just said it's the most humbling day of his life. Too fucking right. Apparently he said it like a dead fish." Has he been hacked? Well, he has his suspicions, but the stories that surprised him weren't in News International tabloids, and he's more interested in the meta-drama than any tiny role he might have played. "There's something Greek about it, isn't there?"

Tennant is still best known as the Doctor. For many he is the Doctor – geeky, manic and a little bit gorgeous. He has also romped as Casanova, probed as DI Carlisle in the TV musical-drama Blackpool, theorised as cerebral scientist Arthur Eddington in Einstein And Eddington (stick a pair of specs on him and he's as dull as the next man), played Hamlet quite beautifully (awkward and paranoid, yet graceful) and appeared in a number of none-too-impressive movies. Perhaps that's the biggest surprise – that for all his popularity and talent, he isn't a huge film star. Maybe this will change with his first Hollywood movie. In Fright Night, a remake of the 1985 original, he plays veteran vampire hunter Peter Vincent like Russell Brand on speed. Sure, he's funny, but it's not the most challenging of roles.

In Much Ado, he is a brilliant Benedick – shallow, laddish and witty, he gradually learns how to show his love. Tennant has a wonderful way of demystifying Shakespearean verse without patronising it, mining every word for maximum humour or pathos. I tell him I loved the fact he looked so chuffed at the end. Well, what could be better, he says – acting in a great play to a sold-out audience with one of your best mates? (Tate also starred with him in Doctor Who.) This is what makes him so likable – you don't feel there is a hidden beef, an itch of resentment that he isn't a global superstar.

It's not that he's unambitious. Anything but. Is it true he knew he wanted to be an actor when he was three years old? He laughs, embarrassed. "I know it's absurd and precocious and, of course, at three, you don't really know what that means. So I was just very fortunate that as I grew up and began to understand what that actually was, the idea developed consistently." He stops, and says actually that's not right – he did have a sense of what acting was back then. "I remember a conversation with my parents about who the people on the TV were, and learning they were actors and they acted out this story and just thinking that was the most fantastic notion, and that's what I want to do. And I remember understanding very clearly the difference between the fantasy and reality of that, and that making it even more exciting."

Tennant was born David John McDonald and grew up with his brother and sister in a manse in Paisley, Scotland. His father Sandy was the local Church of Scotland minister. Would Sandy have liked his son to follow him into the church? "No, no, we all went different ways and that's absolutely as our parents would have had it." If anything, he says, his dad would have fancied a go at Tennant's job. "Being a minister is sort of like acting and my dad has always been very supportive. I think he had a bit of a notion for it himself. He's often said that if he'd grown up in a slightly different place, slightly different circumstances and slightly different time, it's something he might have seen himself doing." His father appeared in a non-speaking role as a footman in Doctor Who.

Young David enjoyed school and worked hard – if he hadn't gone to drama school he would probably have studied English at university. I tell him he was particularly convincing as the sober academic Eddington. Is that a part of him? He smiles. "Oh yeah, yeah, definitely, there's definitely a swot inside me."

His friend Arabella Weir, who was his landlady when he moved to London, once said he had a "steely determination". Does he agree?

"It's funny this quote is always brought up, and I don't feel very steely." What does she mean by it? "I imagine she'd say it's about a determination of purpose, which I can't deny. Certainly, in terms of being an actor, I was very single-minded. That didn't feel 'steely' to me."

As he's talking I take out a couple of old-fashioned cassettes from my bag. "Ooooh look at them!" he squeals excitedly. "From the 1970s!" He's bouncing with delight. At times, Tennant could suffocate you with his enthusiasm.

Now he's thinking about his single-mindedness. "I guess Arabella would say I'm swotty, you see. I make sure I learn my lines… it's just we come from slightly different starting points, I guess." What's his starting point? "Fear, probably. Fear of being found out, fear of not being any good and just, you know, fear of failing."

He looks remarkably tanned for a man who's spending so much time in the theatre. How come? "Oh, this is all fake. We get spray-tanned underneath the stage once a week. I think I look vaguely jaundiced. Paisley people aren't really meant to tan. We're meant to be parchment white."

A few years ago Tennant traced his family's roots in the television series Who Do You Think You Are? It was rather sad, and moving, as he discovered that his liberal credentials weren't quite as impeccable as he'd imagined – it turned out his maternal great-grandparents had been flute-flaunting, Catholic-bating bigots back in Northern Ireland. You could see the hurt on his face as he realised his own family had championed everything he despised. "I was really shocked. It was absurd how naive I had been. Of course, two generations back, I should have realised it was going to be tribal."

Another thing that emerged in the documentary was his tendency to make a bit of a mess of things – such as when he drinks from a stream, says how fresh the water is, only to discover sheep regularly pee in it. Yes, he says proudly, "I've always been a geek and slightly awkward… slightly umm… I was never the cool kid at school."

Isn't it strange, then, that he has become a heart-throb and that women… He stops me before I'm finished. "It's absurd and ridiculous and…" He sounds anxious, defensive even. But isn't there a bit of him that wants to gloat; to tell all the kids who thought he was a nerd that he's now this babe magnet, this sex god, this…

And now he really is flushed and flustered. He wants to get the words out but they won't come. "The moment one is sort of made aware of that sort of thing it feels very… it's very hard to enjoy because it feels so absurd and unconnected to… how do you make use of it, or how do you channel it, or how do you even feel good about it because… because… you're patently aware it's not true."

'We get spray-tanned underneath the stage once a week.' Photograph: Zed Nelson for the Guardian

Blimey. But he's not finished yet. He's determined to stumble towards an explanation of why he's neither desirable nor a conventional leading man. "I mean, it's not true because it's not to do with you, it's to do with characters you play, it's not to do with who you are, or even what you look like, and so I still find it ridiculous when I read a script that says something like 'good looking' or 'fanciable' or anything like that. I think, 'That's not my part' because I always played the weird geeky outsiders. I got cast in Romeo And Juliet in 2000 and people were saying, 'Oooh, unconventional casting.' Then I did Casanova on the telly, which was about a great lover, but that part in that programme he's not – he's a sort of puppy-dog boy."

Hold on a sec. Does the gentleman not protest too much? After all, when he played the Doctor, rarely a week passed without the red-tops suggesting he had a new girlfriend. "And we know we must believe everything we read in the paper!" he says, sardonically.

But did he not get a reputation as Mr Shagtastic? "From whoooooo?" His voice rises a good octave.

"The tabloids suggested you had been out with every woman on the Doctor Who set," I say. He howls with laughter. "Hahahahaha! I refute that heartily." He pauses. "Well, I mean I have had some girlfriends, and I did meet some of them at work you know…" Another pause. "You know, my bedpost really has very few notches compared with other actors of my erm, erm, pedigree. Hehehehehe!"

If anything, he says, he's been Mr Monogamy. "I have never really overdone it. You know, I've never had three on the go at the same time." What about two? "No. No, because I'm too racked with guilt in every corner of my life to even try to get away with something like that."

It's the old Presbyterian thing, he says, just like the work ethic. "It's all connected, isn't it – that sense that you're not worthy and therefore you have to prove your worth, and you don't get above your station."

Does he think religion shaped his character? "I'm sure it must have." He still goes to church occasionally. "There's a morality… I think there's a moral compass but whether that comes from religion or just from being a good person, and where one starts and the other begins…" So what is he? "I'm a good person, I hope. But I'm never as good as I want to be, never as nice as I want to be, never as generous as I want to be."

Tennant brings warmth and likability to most of his roles. Is it important for him to be liked in real life? "I think it's quite important. Actors often have a reputation for being ludicrous and arrogant, and I don't think either are necessary and I think because you produce work collectively it's important to be respectful and receptive, and frankly there's too many of us. It's an overcrowded profession, so there's no excuse for behaving like a twat. And I don't like people who do."

After drama college, Tennant joined the left-wing theatre company 7:84 (the name came from the 1966 statistic that 7% of the UK population owned 84% of the nation's wealth). Was it the politics that attracted him? "Not particularly, it was more that they offered me a job and straight out of drama school that's all you sort of cared about. I mean, it so happened that I agreed with what they stood for, but I would probably have taken a job even if I didn't." How would he describe himself politically? "Left-leaning." Socialist, liberal? "Both of those things." He is a Labour party supporter, did the voiceover for an election broadcast last year and has never had time for the Scottish Nationalists. "I had no great sense of nationalism when I was in Scotland, and I could never understand why the SNP were banging on about it. I was like, why do we want to become smaller? Surely we want to expand and look outward? Let's go into Europe and be one big happy family!"

Tennant is happy to talk about anything but his personal life. And when asked, he deflects with considerable skill. But however much he might resist the notion, there is one remarkable fact about his choice of partner. Five months ago, his girlfriend Georgia Moffett gave birth to their daughter. Georgia Moffett is the daughter of former Doctor Who star Peter Davison. That means Tennant's daughter will have a Doctor as both father and grandfather. How crazy is that? (There have been only 11 Doctors in the history of time travel.)

Tennant smiles. Silence.

"But it is bonkers, isn't it?" Silence.

"Did you actually go out and seek the daughter of a Doctor to mother your child?"

"That's exactly what I did, yes, and there were very few candidates available. It was a limited field." He grins.

"It is unlikely, isn't it?"

"I know, but seriously, anything I say will get picked up."

"Do you ever sit down with Peter Davison and say, 'This is weird?'"

"I'm not giving the Daily Mail a quote. I'm not doing it… Don't do this to me," he pleads. "Mmmmm, I have a very happy family life, very blessed, yeah. I understand there's an interest, and I don't want to feed it."

I give up. "OK," I say, "I'm going to let you…"

"…wriggle," he finishes.

"Am I allowed to ask if Peter Davison is a nice man?"

"Peter is a very nice man."

So we return to the movies. Is this a new stage in his career? No, he says, "I don't think that tactically. I just like to join the jobs up and hope they will be as varied and interesting as possible. So if I can do a film with DreamWorks, then come back and do a little British film, then do something in the West End, then do something for the BBC, that's great." Could Fright Night make him a movie star? "I suppose it's the first big Hollywood thing I've done and people say, is this going to be your move to Hollywood? Well, of course it's not going to be. You learn from experience that the things you think are going to change your life probably won't." He stops. "Not that I'm looking for it to be changed. I'm very happy with what I've got."

• Fright Night is released on 2 September.