'I'm 40! I'm an adult!" shouts Jude Law. "Aren't I?" We hold these truths to be self-evident, I reply, as the actor, laughing, stares across the table with those adorable baby blues and more hair than's fair. "But," he says more quietly, "part of me thinks I can't play a doctor. Who would come to me?"

You've got to be kidding. Who wouldn't come to Dr Jude? In Steven Soderbergh's film Side Effects, Law plays an Englishman in New York, a slimy limey of a pill-dispensing psychiatrist who becomes entangled in murder, drug switcheroos, a risible lesbian insider trading scam and lots more vaguely voguish, putatively Hitchkockian hokum before the credits. Astute critics have compared this performance with the one Law gave in the 2004 film I Heart Huckabees, where he played a shallow business exec in psychic meltdown. "The de-smugging of Jude Law is yet again a dramatic motor to swear by," wrote the Daily Telegraph. Quite so: seeing Dr Jude losing his Brit cool when wrong-footed by faux-innocent Rooney Mara or handbagged by crackers shrink Catherine Zeta-Jones is worth the price of admission alone.

We're sitting in a conference room at the Guardian in London's Kings Cross. For an hour his PR chaperones have left him alone with the clown who once inadvertently cycled into the canal we can see from the window. If Law sacks his minders later, that would be understandable.

Between us is a pill bottle whose label says it contains 20mg capsules of a new antidepressant called Ablixa. Perhaps if the media inquisition gets too much, you could neck some, even if the directions explicitly state "Take ONE daily" and warn that side-effects include sleepwalking and insomnia. "But they look like Smarties!" says Law. That's because they are Smarties. The PR people for Side Effects, which deals with the perils of prescription drugs, handed out the bottles at the press screening the night before. Nice gag. One could barely concentrate on the final credits for the rattling of antidepressants as the hacks scrambled out of the cinema.

"Are you sure they're Smarties?" asks Law, examining them. That's what the PR person said. "Maybe give them to the kids and then see how they feel." Yes, indeed, that would be the responsible thing to do, Dr Jude. If you wanted your medical licence shredded.

Like Tigger, Law seems in a jaunty mood today. "On the whole I'm happy. What's not to be well adjusted about? I tell you what helps me – running. I've always liked to have a couple of fags and a few drinks and look after myself – the best of both worlds, you know? But recently I've really got into running – it clears my brain."

With Sienna Miller, 2010. Photograph: Matt Baron/BEI/Rex Features

Let's unbounce Tigger with some questions about that most dismal of subjects, media ethics. What is your life like post-Leveson, now that phone-hacking is history and some of Murdoch's evil henchpeople have been reined in? "My life is much less harassed, thank goodness," says Law. How come? "Because the the root has been cut off so it doesn't feed the poisonous little plant that was growing, do you know what I mean? Without headlines they've no need for ridiculous photos of me opening a gate and getting into a car. So that's all gone. What a blessing!"

It's just over a year since the actor accepted damages of £130,000 plus legal costs from Rupert Murdoch's News Corp for its repeated hacking of his phone between 2003 and 2006. Initially, Law suspected friends and family of leaking personal details about his personal life to the tabloids. He changed phones repeatedly, hired security consultants who swept his home and car for bugs, but still the stories about his private life kept appearing. Only in 2011, when police replayed him vociemails left by his children's nanny, did Law realise the extent and nature of the intrusion – what his lawyer called "a sustained campaign of surveillance, pursuit and harassment".

But the legal profession and the police didn't come out of this smelling of roses either. Law told the Observer in 2003 that when he was living in Primrose Hill with his then wife, actor Sadie Frost, and their children, the police betrayed them. "There were two instances where the police called for whatever reason and they sold the stories, telling lies." In that year, too, the affadavit of his decree nisi ending his marriage with Frost was sent from the high court to a tabloid before it reached him.

Nor, though, did Law play a blinder during those phone-hacking years. In 2005, it was revealed that he was having an affair with his children's nanny, Daisy Wright, while engaged to actor Sienna Miller. His by then ex-wife Frost reportedly sacked Wright after one of their children caught the nanny and Law in bed together. Later Wright sold her story. "It was mind- blowing rampant sex," she told the Sunday Mirror. "He is a great lover and he knows how to satisfy a woman." No doubt, but that press coverage prompted his mea culpa to Miller. "Following the reports in today's papers," he said in a statement in July 2005, "I just want to say I am deeply ashamed and upset that I've hurt Sienna and the people most close to us."

Jude Law in Side Effects. Photograph: Sportsphoto Ltd/Allstar

Law in those years certainly wasn't mere victim. "Don't tell me there isn't anyone who has done things they regret, or done things they shouldn't have," he said a couple of years ago. "Or done things that are silly. Or said silly things. That's life, right? That's what's wonderful about life. We all do this stuff we shouldn't do."

Today Law declines to talk about those wild Primrose Hill years. Instead he gives an appealing performance of an older, wiser, more humble Tigger – and one who also feels more hopeful than he has for many years about the power of the press to ruin his life. "What's going to be interesting, post-Leveson, is how long the head stays in the shell, how long before it pops out again. I don't know, we'll see."

But has the root really been cut off? Arguably, the poisonous little plant is still growing. Jude may want to be obscure, but that seems an unrealisable dream. Just before Christmas he was papped walking down the street in Primrose Hill, going to a Greek restaurant with two of his children, their faces pixellated. A few weeks later, at the end of December, he had a 40th birthday party in Los Angeles. The Daily Mail joined the party mood with a story, headlined: "Midlife crisis? Jude Law turns 40 in Los Angeles – and new love is 26." The story began: "Most men at risk of midlife crisis treat themselves to a flash new sports car - but not Jude Law." Yeah, right, "most men": during my many midlife crises I have scarcely treated myself to a second eclair.

On New Year's Eve, Law and his sons were papped body-boarding in Hawaii. In the published photos, his son's faces were again pixellated. "It's a very hard thing to talk about. There's an awful lot of people who think: 'You work in films, what do you expect? Stop bitching and moaning about it.' I'm maybe still blind to the fact that if I go to certain places this is going to happen and I have no right to get pissed off. I feel I do because I'm on holiday with my kids – leave me alone, leave them alone.

"We left the beach because we couldn't play there because there were three guys sitting there taking photos of us. So they can pixellate all they want, but the truth of the matter is that these greaseballs are still there and we have to leave the beach and go and sit in a garden or a room or whatever."

Jude Law in Alfie, 2004. Photograph: Paramount/Everett/Rex Featur

Law pours some water and looks cross. "But look, I really don't want to talk about it. I'm so bored of myself going on about it. And I'm so aware it's such an association with me. I know people are bored with it. I want to move on. I've got to make the decision where I've got to be a lot more private at times – and if I don't like it." At one point you considered leaving London for good? "Yeah, yeah. But I like living in London. I don't want to surround my kids with security, don't want to take my kids to private beaches on private holidays. But maybe I have to. Otherwise you'll have to put up with me bitching and moaning endlessly."

But Law has a problem today. He is giving an hour of his time to the media and thereby feeding what Peter Andre called the "insania". "The predicament I'm in is that when you're in a film like Side Effects which doesn't have a big sales budget and is up against, as ever, the big, more commercial movies, the way to get it out there is to talk about it and that inevitably is going to feed your relationship with the media. So it's a tricky situation." Can I get a boo-hoo?

It is a predicament and not one all creative types have to endure. Later this month (11 March) we will hear Law on BBC4 narrating a film about the late, great Scottish sculptor William Turnbull, made by the artist's son and Law's friend, Alex Turnbull, who is best known as a member of post-punk funk band 23 Skidoo. "Bill made a contribution without having to have a relationship with the media. He dragged British art kicking and screaming into modernism after the war." Indeed: Turnbull's sculptures and paintings spoke for themselves, so Turnbull could remain silent. Law wishes he could be more like that. "In the end the work keeps you interesting and interested. It's real, isn't it? It's a real contribution. That's what Bill did. All the rest – there are those who choose to make the other stuff a part of their contribution. I personally don't, but you get pulled into that stuff whether you want to or not."

Surely, though, you never wanted to be obscure. Isn't narcissistic display part of the allure of becoming an actor? "I wanted to be recognised as an actor, not for my looks or whatever." And very quickly he was, though the looks probably didn't harm his cause. Born in south-east London in 1973, Law joined the National Youth Music Theatre aged 12 and filmed his first TV series, Families, aged 17. In the early 90s the tyro thesp did a lot of theatre, with directors such as Katie Mitchell, Matthew Warchus and Sean Mathias in productions of Death of a Salesman, Les Parents Terribles and Indiscretion. And then, aged 23, two things happened: he became a father and his film career took off.

Jude Law in the The Talented Mr Ripley. Photograph: Sportsphoto Ltd/Allstar

"I'd done 10 or 12 plays back to back. Suddenly this new medium offered itself and I was fortunate enough to work with some good people. Plus the money helped with a young family. It felt like an exciting time." In quick succession he was Oscar Wilde's lover opposite Stephen Fry, then a disabled former swimming star in Gattaca, then a hustler murdered by Kevin Spacey in Clint Eastwood's Midnight in the Garden of Good and Evil. Before he was 30 he had been Oscar-nominated twice – for his performances in The Talented Mr Ripley and Cold Mountain, both directed by Anthony Minghella. True, he did star in the last film I walked out on, Enemy at the Gates, in 2002, but let's not spoil the story: by 2006 he was in the top 10 of bankable Hollywood stars.

"I never planned a career in film. I don't know anyone who grew up in the 70s in Lewisham who did. It just wasn't realistic. Theatre was. I grew up watching and loving film but it was this other world – movie stars were like Newman and Mitchum and McQueen. And then this amazing crop of actors – Tim Roth, Gary Oldman and Daniel Day-Lewis – came along. Tim and Gary were round the corner from where I lived. That was very, very, very influential for me. I remember seeing Daniel Day-Lewis in My Beautiful Launderette and thinking: 'Oh, film can be part of my world as well.'"

By 2006, the London boy turned Hollywood A-lister had misgivings about his career. "I had to take a handle on what I was doing. I had to make a bit of money for the family [by then he was father to three children with ex-wife Sadie Frost] but I also had to think about how to please myself – and working in a play every year or every two years was an important step in that direction." Why? "I'll be honest. I feel – oh, I'm not going to give you that cliche that I feel more at home – but I feel more in control of the process in theatre and it's more familiar to me."

He returned to the stage in May 2009 as Hamlet. "I wanted to play him before I was 40. My feeling is always commit and do it. You don't want to get to 50 and have not played Hamlet." In the autumn we will see Law as Henry V, again directed by Michael Grandage. "I wanted to play Henry before I was 40 too, but I just missed it. He died when he was 37 so he has got to be played as a young king and I think I can get there with a little help from prosthetics. I'm joking."

Law doesn't yet do prosthetics. He does, though, bulk up. When he played Mat in the Donmar production of O'Neill's Anna Christie he had shape-shifted: he was chunky, lavishly bearded and, most improbably, Irish. "I was playing this guy from the west coast of Ireland so the accent was really tricky and really thick. And I was eating a lot of weird food and exercising to bulk up. I remember halfway through rehearsals wondering 'What am I doing? Am I in a pantomime? They're going to kill me!'" Who – the critics? "Everyone! Me! I'm going to kill me for doing this."

His performance opposite Ruth Wilson in Anna Christie in 2010 was critically feted. The Guardian's Michael Billington wrote: "I suspect it's a breakthrough performance in that it releases Law from the tyranny of always being seen as the good-looking lead man and allows him to become a character actor." As if to prove that point, the following year we saw Law in Joe Wright's adaptation of Tolstoy's Anna Karenina, as balding non-hottie Karenin who, in an emblematic scene, takes a silver case out of his bedside table containing his reusable condom for cheerless sex with his understandably adultery-hungry wife (Keira Knightley). Soon, too, we will see him as the eponymous Dom Hemingway in a Brit flick in which he plays a chubby hood in nasty synthetic shmutter and a heinous beard.

You were reported as saying that this was going to be your decade. "I would never say that! Jesus God no way! I said I was optimistic about my 40s. The roles should get more complex. I look back and I'm proud of the work I've done but not fulfilled by it. I feel in some areas I've only scratched the surface." One itch he plans to scratch after Henry V is directing. His last time on the other side of the camera was in 1999 (on Tube Tales). Now Soderbergh has, reportedly, made his last film, perhaps you could take his place. "I wouldn't presume! But I do want to direct and I've got a novel I'm developing." There's a tantalising possibility that he and Werner Herzog might work together. "He rang up and said, pretty much: 'You, me, Borneo, piano, river, camera.' And I'm like 'Werner, I'm there.'" The money isn't yet, though. Shame: I'd pay to watch Jude Law as a latter-day Klaus "Fitzcarraldo" Kinski going nuts in the tropics. Make it happen, Hollywood.