He is a famous comedian, but there is nothing funny about Billy Connolly's latest role: he plays a tortured Catholic priest in The X-Files: I Want To Believe. "There are no laughs at all, which is brilliant," says Connolly gleefully. "Father Joe is very disturbed, very dark. Doing a drama like this is a joy; you get to use your body in a different way, you get to use your eyes. I love playing dark characters, the darker the better, the more disturbed he is, the happier I get." He fixes me with menacing eyes. "The most worrying thing is that the darker the character, the easier I find it. If he's a really nice, intelligent man, I find it awfully difficult."

Connolly has tackled drama before, notably in the film Mrs Brown, with Dame Judi Dench, but he's never portrayed anyone like Father Joe, who is psychic and possibly deranged. "I was brought up as a Catholic," Connolly says. "I have several relatives and pals at school who became Catholic priests – aye – I have a cousin who is a nun and another cousin who is a missionary priest in Pakistan." He pauses and smiles. "And I am an atheist."

We're meeting for tea, close to the X-Files set. Connolly is dressed in black for the part, his hair white and wild, his beard long and shaggy. "I'm sure everywhere I go people think I look like a wino and go, 'Look at him, is he OK?'" he says. It seems to bother him. "I have to go around like this all the time and it's kind of weird and unpleasant. But this guy is great, completely crazy," he says. "I love playing people who are spiritually adrift. If the characters are sociopathic or psychopathic, that's fantastic. I love playing people who are capable of anything. If I go to the lengths of filming in Vancouver for three months, away from my family, I should have a good reason; I can't imagine anything worse than being in a big summer comedy about food fights in college and stuff like that."

The X-Files film is the polar opposite of a lightweight, sunny blockbuster. Six years after the popular sci-fi show ended, David Duchovny and Gillian Anderson are back as the FBI agents Fox Mulder and Dana Scully, investigating the apparently inexplicable abduction of a woman in the mountains of Virginia. Conditions are treacherous, temperatures sub-zero.

The director, Chris Carter, who created the series, promises a chilling and grimly terrifying story. All he will reveal is that there is some kind of paranormal story line, but no alien abductions – and he has banned his stars from giving away plot details. "I think the secrecy's great," says Connolly. "But there's also something I quite admire about the anarchic side of getting hold of a secret and exploding it." He laughs. "If the secret gets out it won't be through me, though."

Carter wrote the part of Father Joe especially for Connolly. "I would say I was only mildly interested in the TV show," Connolly says. "But my daughter Amy is one of the X-philes; she can't believe I actually get to talk to Mulder and Scully. I couldn't tell her anything about the film, though. I told Pamela [Stephenson], but she's an exceptional case, she's a shrink." He's referring to his wife of 19 years, the former Not the Nine O'Clock News star, who is now a clinical psychologist and writer.

Connolly is always cracking up, which adds to his charm. The chemistry of his expletive-laden, expressive language, combined with his physical presence, is compelling. He's like a hyperactive child, full of boundless energy. While many comedians can be quiet, even morose, he seems happy to talk about himself and his work.

His co-star Duchovny tells me that Connolly keeps cast and crew entertained between takes. Not a method actor, then? "No," Connolly says in mock horror. "I don't stay in character and I don't like people who do. I think it's pretentious and self-indulgent. I think if you need to stay in character for the whole movie, you should consider what you're doing for a living. I know people who do it and they're awfully good, but it's not my cup of tea."

Now 65, he's appeared in many films, from The Last Samurai to Lemony Snicket's A Series of Unfortunate Events. Many have been commercial flops, but he doesn't seem to mind. "I think they've all been great, whether it was Fido, The Man Who Sued God, The Debt Collector or The Boondock Saints. Some were small films that didn't make it, for whatever reasons that are beyond me, but I'm immensely proud of them. Of course it's nice when a lot of people see them – they will see this one, obviously – but I don't have a problem with films being overlooked, either. If you look at my film record, right back to Absolution with Richard Burton, they stand up."

Mrs Brown is his favourite. "I loved that one. It was on last Christmas and I watched it right through. People were shaking my hand on the street even though it's been out for years. Then I was on Michael Parkinson's last show and they showed a clip from the film of me and Judi shouting at each other. I was so moved, my lips started to quiver. It was amazing."

Which part of his career does he prefer? "I like dividing my life between comedy and drama," he says. "Usually, for some reason known best to someone else, I'll get a film offer after I've been on the road doing the funny stuff. I love the discipline of being in a film. Another set of rules apply, where I have to say exact words at a certain time so another person can say the next thing – that's not required of me when I'm doing my comedy; I can say it differently every night.

"In films you have to think of other people, whereas in my comedy life I only think of me, no one else, I'm up there completely alone on the stage and what I say goes. It's liberating, it's exhilarating, but it can entrap you. You can become a Hitler, because thousands of people every night are telling you how great you are and you can get carried away sometimes. Usually you don't notice it until you come off the road and start shouting at your wife – 'Where's my tea?' It can be quite distressing. Doing films makes me more human, because if I stay out there on the road, I become such an unbearable bore, shouting and getting my own way. You slip into these things. I'm not saying I'm a bully or anything, but I'm very used to getting my own way."

Clearly, Stephenson does not tolerate her husband's dictatorial demands. "With Pam, I discovered that you could not get away with anything. When I married her I thought, 'Oh God,' because I had to own up to everything, which no one had ever asked me to do before. I learnt to be honest with myself, which was great.

"And she helped me make positive changes in my life. I was a bit of a drunk and she put me right on a few things. Self-medicating – that's usually the working-class answer to everything, throw a few stiff ones down you and that should sort it out, the problem will go away."

With the help of his wife – and therapy – he has been sober for decades. Connolly and Stephenson have three daughters, Daisy, 24, Amy, 22, and Scarlett, 19. (He has two older children, Jamie and Cara, from his first marriage.) They live in Manhattan and have a house in Aberdeenshire. "I'm a great family guy, I love them all," he beams. "I spend all my money, so I have to work. But I have a good balance – I go fishing, go out on my motorcycle, I read a lot and watch telly. I love spending time with my girls. They were raised in Los Angeles but we moved to New York because they go to college on the East Coast and if I lived in LA I would hardly see them."

Connolly's own upbringing was tough. He grew up poor, in a Glasgow tenement; his mother Mamie abandoned the family (he has an older sister, Florence), and they were raised by his abusive father, William, and aunts Mona and Margaret, who beat him. Yet he doesn't believe his comedic talent was the result of poverty and suffering.

"Your background has little to do with anything. It doesn't dictate what you're going to do and I don't see myself as a victim of poverty, or anything like that. I think comedy comes from darkness, from inabilities. All the great comedians have been unable to do things – or they do things badly. Look at Charlie Chaplin and Laurel and Hardy. There were comics like Dean Martin, who were good at doing things, but they're few and far between. Great comedy is all about telling real things, the truth, in a light way, so the audience goes, 'God, that's right, I felt that,' and they burst out laughing. There's a lot of truth involved that makes people identify with you."

What strikes me about Connolly – beyond the jokes – is his sheer enthusiasm for life, which is infectious and quite different from the stereotype of the angst-ridden comedian. "My life is great," he says, "because people are always happy to see me, so my image of the world is happy. Some people think the world is a terrible place, but I think they have got it wrong. I'm very, very optimistic."

Clearly contented, the man who started out as a welder in the Glasgow shipyards, before forming his own folk group, The Humblebums, confesses that there is another challenge he would like to pursue. "I don't play music for a living any more, but I still play banjo with my pals – Steve Martin, guys like that. I found a wee pub in New York where they have old-time banjo and fiddle on Wednesday nights, so I've been along, but I'm kind of nervous. I want to play with them, but they're a bit better than me. I have had to ask myself, 'Why am I doing this when some other guy can tear his ass off better then me?' It is sometimes good to do something just because it's difficult. So I might have a go."