Billy Postlethwaite has lost his matches. There’s a hole in his pocket, he mumbles, shaking each leg in a little jig. Eventually, the box slips on to the rehearsal room floor. Snatching it up, he flicks back his mop of curls and grins. He doesn’t half look like his dad, especially when his eyes flash with mischief and grow ever wider. This happens when we talk about two great men of theatre: William Shakespeare and Peter William Postlethwaite.

For Billy, the pair are entwined. Most British kids encounter Shakespeare at school; not many find their dad reeling off soliloquies over their cereal. When Postlethwaite Sr was playing Prospero, Billy’s sister, Lily, ordered him to stop calling her Miranda at breakfast. In his memoir, he recalled becoming so obsessed with Macbeth that he answered innocuous questions from his children with tortured lines from the tragedy.

Now Billy is playing the murderous Scot at the Watermill in Newbury, Berkshire. The company are staying in cottages next to the theatre, which is a converted mill, and we walk over to the house of artistic director Paul Hart. Dogs greet us by the kitchen. Waddling outside are muscovy ducks, which don’t quack so won’t distract audiences.

A minute’s stroll from bed to rehearsals beats commuting, says Postlethwaite, who grew up in Shropshire and calls himself a “proper country bumpkin”. Looking out of the window, he says: “I swim in the river and in that pond by the old mill.” Hart responds: “I am worried you’re going to die of hypothermia.” The other actors join these early morning wake-up swims. “Lady Macbeth was the first one in,” cackles Postlethwaite. Is the water as cold as it looks? “Bloody freezing!”

This is Postlethwaite’s biggest Shakespearean role to date. His stage debut was as the lovesick Silvius in As You Like It, opposite Cush Jumbo, in Manchester in 2011. Two years later, he was Edgar in director Lucy Bailey’s 1960s-set King Lear, with its East End gangland empire.

“The plays affect you in different ways,” he says. “In As You Like It, you have to look after your heart because you can fall in love very easily. I did A Midsummer Night’s Dream in Bath, and we used to drink and go out an awful lot. The spirits and fairies send you off doing stuff you shouldn’t really do. Macbeth apparently sends you crazy.” He gives Hart a look: “Which you kind of want and don’t want, don’t you, Paul?”

He was eight when he saw his dad’s Macbeth. “Apparently, I fell asleep,” he says, roaring with laughter. Actors often avoid watching others play a role they’re learning – let alone a celebrated performance by their own father – so I’m surprised to hear he has hunted down a recording of the 1997 production. After The Usual Suspects, Brassed Off and Amistad, his dad was at the height of his fame and turned down the offer of Saving Private Ryan to tour Shakespeare’s tragedy. “Of all the tortured souls I’ve played,” he said at the time, “no one understood the prickly heat of guilt like Macca.”

Postlethwaite recognises that, turning 30 this year, he’s younger than most Macbeths. He says a Scottish friend gave him two pieces of advice: “Don’t do it in Scottish and remember it’s a bloody romance.” The murder of King Duncan becomes, in this staging, almost a “loving act” for the Macbeths. He wants to tap into the play’s “weird sense of fun” and says the intimacy of the Watermill means it can be a chamber piece.

This Macbeth has its own house band in the form of the “weird sisters” who will crank out tunes including The House of the Rising Sun and Intro by the xx. They’re hoping to find a place for Trent Reznor’s Hurt, whose “empire of dirt” and “crown of shit” should feel right at home. Hart says music has been key to the Watermill’s recent Shakespeare productions, both in appealing to audiences unfamiliar with the plays and as a shortcut to the subconscious of the characters. The witches will supply various sound effects. “Every sound they make is something only Macbeth can hear,” explains Hart. They are “instruments of darkness”, adds Postlethwaite. “They facilitate dark deeds.”

Hart and Postlethwaite worked together on Journey’s End at the Watermill in 2014. “We’ve been looking for something else to do together but Billy’s got quite massive on telly,” says Hart. “That’s absolute bollocks,” Postlethwaite protests. But telly doesn’t get more massive than Game of Thrones, even though it took just a day for Postlethwaite to film his fleeting appearance as a Lannister soldier in one episode. He shares a song with Ed Sheeran in a charming campfire scene that also features Thomas Turgoose. The trio play soldiers who share roast rabbit and blackberry wine with Arya, played by Maisie Williams.

“I’ve been told it’s one of the most hated scenes in Game of Thrones,” Postlethwaite says with a laugh. But his character had a “lovely little story to him – this soldier who is away from home, has just had his first baby, and doesn’t yet know if it’s a boy or a girl”. A fan-fiction plot suggested his child would be killed by the queen, leading Postlethwaite’s soldier to join forces with Arya. “I was like, ‘What? Send that to the producers!’”

His biggest role on British TV was as unhinged registrar Fredrik Johanssen on Holby City. Fans were enraged when he killed off newlywed surgeon Raf (Strictly Come Dancing winner Joe McFadden) in a shooting spree. Postlethwaite’s films have veered from the blockbuster Tomb Raider reboot to the tense, low-budget Containment, in which he and Louise Brealey play a couple trapped inside their high-rise home while mysterious figures in hazmat suits swarm outside. Postlethwaite’s character is a sneaky piece of work, ready to forsake other residents to save his skin.

“I do seem to play these sorts of snakey characters,” he admits, “but they can be quite fun.” His father was always asked to play villains. Does he worry about typecasting? “Beggars can’t be choosers,” he replies. “I like to find the light and the dark in everybody. It’s what Dad managed to do, highlight villainous aspects of people but also great compassion, love and care.”

When his father was at the Bristol Old Vic Theatre School, a tutor told him he had a face “like a fucking stone archway”. Billy says when he got turned down for a part himself, his agent told him: “They said you were too ugly. But it’s good because it means you’re a character actor and you can play loads of things.” He mulled it over: “I’m a character actor for 20% of my life and the rest of the time I have to walk around with this ugly mug!”

The way he tells it, pursuing a career in acting had as much to do with the number of girls at his drama club in Shropshire as his dad’s career. He arrived at London Academy of Music and Dramatic Art (Lamda) in 2008 and was soon blown away by Mark Rylance’s performance as Rooster, ruler of the Wiltshire woods, in Jerusalem at the Royal Court. “That rave outside in the countryside?” he says. “I was like, ‘I’ve been there. I know this.’” His dad was “amazingly helpful” during drama school. “I’d ring him up and say I’ve got no idea what I’m doing. He’d say just one word or a sentence and it would unlock things.”

Pete Postlethwaite died in January 2011 during Billy’s final year at Lamda. He left school early to do As You Like It in Manchester then had roles at the National and the Young Vic. Florian Zeller’s typically elusive The Mother, staged in Bath in 2015, cast him as the son of a desperate mum played by Gina McKee. He asked his own mother what the most painful part of parenthood was. “She said, ‘Letting go.’ It was the most helpful thing.”

Now here he is doing Macbeth, roughly 20 years after his father, in what sounds like his own letting go. “I said I’m going to come and raise – and put to bed – demons,” he says. “Who am I as a person? What am I doing acting? Am I just trying to copy my dad? Am I my dad? Am I not my dad? All of these dark, horrible thoughts.”

It’s time, he says, to “put all that stuff down and kill it and go, ‘Well, now be you.’” Again, the eyes flash with mischief. “While being Macbeth!”

• Macbeth is at the Watermill theatre, Newbury, until 30 March.