Charles Voyde Harrelson - enforcer, gambler, and man of education and charm - was first convicted of killing for hire when Woody was seven. Ten years later, in Texas, he was imprisoned for murdering federal judge John H Wood in what the FBI described as "the crime of the century". The father protests his innocence. The son believes it while acknowledging that Harrel-son senior has done "a lot of illegal stuff". As if anything were lacking from the story, he reportedly also confessed to having killed John F Ken-nedy, though he later withdrew that statement.
Charles Harrelson walked out on his family in 1968, but in adulthood Woody has rebuilt the relationship. "I don't see him as a murderer. I see him as Dad."
He's running late, but that's par for the course at a film festival - Venice - for an actor who spent most of the night in the hotel bar. He's drinking what looks like lemon juice now. "I'm definitely in the low gear," he complains, grabbing at a buzzing fly. "One more hour's sleep and I'd have caught it." His voice has a Texan twang.
At 33, he's less substantial in the flesh than he appears on the screen. Blond, with blue eyes not too close together; friendly, though edgy. But anyone in his position would be. Personal stuff apart, NBK (as the cast call it) caused a furore. Harrelson plays one of a pair of lovers on a killing spree, made into heroes by the (fictional) media. It has been suggested that the film provided a role model for 10 real-life killings in the States. Though the links were later shown to be unfounded, that led to a delay in the film's being given a British certificate. And concern remains not only about the sheer scale of the film's violence but as to whether Stone himself isn't guilty of glamorising his protagonists.
Echoing (or quoting) Stone, Harrelson claims the film's message, on the contrary, is that "love conquers the demon". He pronounces it "lurve". He says the film reflects the "chaos and madness" of young people's lives and claims to see "redemptive qualities" in the finished product which were missing from Quentin Tarantino's original script. (Tarantino later disowned the film, claiming that his screenplay had been altered beyond recognition.)
Life on the set, he says, "was like an MTV video. Oliver would jam on the music, the lights would be wild, sometimes they'd shoot shotgun blasts to get up the energy. In a movie about violence you have to show what you're attacking."
His character, Mickey, "is on a collision course with his own psyche". It's tempting to say the same of Harrelson. "My life has been this really strong push-pull," he once said. "I'm grappling between myself and my shadows." Professionally, from the sweet bar-mascot of Cheers onwards, his work shows a progression of which NBK is the logical conclusion.
He joined Cheers in 1985 and the part later won him an Emmy. At the same time he was pressing forward with a stage career, and it wasn't long before he was finding movie work too.
In Doc Hollywood, he was just a hick to Michael J Fox's city slicker, but in the basketball movie White Men Can't Jump he was half conman and half mark, playing off his own dopiness. The duality was used more strikingly in Indecent Proposal, where he sold his wife's body to Robert Redford and still managed to emerge as the romantic hero. But there was - rightly for the part - something weird about him in that film. Even playing a professional, an architect, "the white-trash element" in Harrelson (as Oliver Stone calls it) comes through very clearly.
Harrelson was raised in Houston and Ohio by his mother, a devoutly Presbyterian legal secretary. Diagnosed as hyperactive, dyslexic and emotionally disturbed, he was sent to a school for slow learners and prescribed the controversial drug Ritalin. He learnt to act in high school. He learnt to loosen up in college, where he arrived as a Bible-basher studying theology and left to look for work on Broadway.
But strange stories cluster about his family. Last autumn, for instance, a step-sister he was about to meet for the first time disappeared off a shrimp boat in the Gulf of Mexico, amid suggestions she may have faked her own death to leave an abusive boyfriend. Harrelson himself once said he felt trapped in a relationship after more than three hours. A chain of girlfriends included Glenn Close, Carol Kane, Moon Zappa (daughter of Frank) and Brooke Shields, and there was a brief marriage to Neil Simon's daughter Nancy. Today he has a toddler by his former personal assistant Laura Louie.
He's a vegan who openly advocates recreational drug use. A practitioner of yoga, an environmentalist, a writer of poetry, a New Age teepee-dweller and a sitter on mountain tops in search of visions. Literally. He took a stand against the Gulf war, at real risk to his career, and played in a band called Urban Masaya. Rather than the fashionable Buddhism, he studies Hindu. "What I love is that there's destruction and then regeneration, know what I mean?"
He works on the elimination of desire - for women, for money - and practises non-ejaculatory Tantric sex, supposed to preserve energy. "I wasn't coming for three months making NBK." It would be easy to make him out an amiable fruitcake but his rapid rise suggests an element of toughness, and of calculation too.
"We have a 10-minute conversation and you say, `Oh, he's so open, so honest'. But I have definite walls at a certain point. I've had a lot of violence in my life. I've been in a lot of fights and had a lot of rage. I was hurt as a child, I guess a lot of us are, and rage is related to fear."
There is, Oliver Stone says, "a degree of violence that is buried in Woody. You see it in the eyes. He's not a goofball sweet charming guy all the way. Anybody who does yoga for sometimes eight hours a day has got to be suppressing something." A valid point. But it's only fair to say, by way of a retort, that Harrelson claims his NBK character is modelled on Oliver Stone.
! `Natural Born Killers' (18): Odeon West End (0426 915574) and nationwide from Friday.