Brendan Gleeson, master of the flawed figure of power, is preparing for shockwaves with his new film Calvary – about penance and paedophile priests. The ex-maths teacher talks to Catherine Shoard about faith, impending doom and beautiful deaths

There's apocalypse in the air the day I go to Dublin. In the driving wind and rain, traffic seizes up, umbrellas crumble, and my map turns to pulp. Later, planes won't land and lights won't work. Thank heavens, then, for Brendan Gleeson, holed up in the basement of a hotel near parliament. He grips my hand and grins, massive and unflappable, as the courtyard outside the window seems to explode.

For all his smiling, though, it seems Gleeson isn't in the most reassuring mood: the apocalypse isn't just swirling around Dublin. "The whole world is in cataclysmic disillusionment," he says, pouring his fizzy water. "There's a sense of grieving going on for the loss of clarity over what's acceptable and what's not. People don't trust in the concept of goodness, or in the authorities defining it for us. Religion in general has been dismantled in western Europe. All systems – socialist or capitalist – are crashing."

The time is right, in other words, for Calvary, Gleeson's new film, premiered the night before for the home crowd. "There seemed to be a pleasant sense of shock in the room," he says. "A pleasant whiff of trauma. The reaction tends to be muted – you don't get standing ovations. But you know it's working."

Calvary is the first film to face up to the sexual-abuse scandals that have so tainted the Catholic church lately. Its genius lies in doing this upside down – by querying the anger as much as channelling it. Gleeson plays Father James, a good priest in a west-coast town riven with sin and spite. In the first scene, he takes confession from a man who's known to him but anonymous to the audience. The man tells him he was repeatedly raped as a boy by a priest, who's since died. So next Sunday, he's going to murder blameless Father James as an enforced act of penance. "I'm going to kill you, Father," says the man, "because you've done nothing wrong."

The film is the second in what director John Michael McDonagh calls "the glorified suicide trilogy". This began with 2011's The Guard, a brilliant comedy starring Gleeson as a Connemara cop with a fondness for class-A drugs and prostitutes, and will conclude with a film in which he plays a paraplegic who hates the able-bodied. Like The Guard, Calvary is tartly, tightly scripted; unlike it, it's a pious piece of work, a serious investigation of expiation.

Gleeson is in almost every scene, but less gabby and fiery than usual; he's the still centre around whom flashier names – Dylan Moran, Aiden Gillen, Chris O'Dowd – spark and spin. Their characters, says Gleeson, one of whom may be the one planning murder, "pour their own cynicism like a vial of poison into James's ear. They want to show him up as a hypocrite, if that's what he is. But at the back of it, there's this lingering hope he won't break."

Really? They seem pretty intent. No, says Gleeson firmly, adding that nobody wants to end up disillusioned. "I think they settle for it sometimes and they punish somebody who gives them an alternative – but they still want to be persuaded." So people just want a pound of flesh? A careful pause. "I think it would be acceptable for a pound of flesh to be extracted. But people are aware of what that costs, too. If you have a revolution, you need to be damn sure what you're replacing the old order with. Not just tearing things down without a viable rebuilding programme.

"What people are excited about is that this dilemma has been aired. I think it's what art should do: make you feel less alone – either in the quest for truth or in dealing with any pain you have. Everybody had a private reaction to it last night, but there was a communal catharsis. It's out there, which is a huge relief to everyone."

When he was 18, Gleeson went to France and was shocked by its secularism. "We'd come from a place where the church would be full for four or five masses on a Sunday morning to a place where nobody went. Well, maybe three or four elderly people." Now, he thinks, Ireland is playing catchup, and the time is nigh to start imagining a post-religion society.

The elephant in the room is Gleeson's own faith, which is off-limits today. He won't even give a verdict on the new Pope. Why not? "Anything I say at the moment will damage the film," he says. Should he come out as a practising Catholic, he thinks people will read the film as a defence of the church; if he says he's renounced religion, it'll be seen as an attack.

At first, this feels slightly confounding. So central is the theme to the film, all subsequent questions feel leading, intentionally or not. He was certainly raised devout, the son of Pat and the "hugely religious" Frank. He attended Catholic school in Dublin and taught maths until he was 34, then quit to act full-time. His wife of 32 years, Mary, is a welfare officer; they have four sons, two of whom have followed him into acting. Every opinion he ventures – about suicide, society, on-set behaviour – suggests a liberal socialism rooted in scripture and reconciled to reality. He fairly bleeds reason and wisdom.

Do people ever treat him like a priest? "No! Yes! I can't say," he laughs. Yet there is something of the perfect preacher about Gleeson. Today, he's clad totally in black, clergy-style, layer upon layer of linen from his ankles to his collar. A faint dog-collar effect is lent by that all-white chin, the rest of his rusty beard creeping over his cheeks like a delightful kind of lichen. He's projected guts and commitment in a career peppered with leaders of all stripes, from Michael Collins to Winston Churchill, from Martin Cahill to King Menelaus, and from Monk the sheriff in Gangs of New York to rock-lobbing Hamish in Braveheart. His characters often die, but even the really criminal, like the In Bruges hitman Ken, tend to do so as martyrs. And, for all his foibles, Mad-Eye Moody from Harry Potter is a man you'd follow, too.

Now aged 59, he's devoted thought to pastoral matters. When working on a film, he says, "it's incredible how much more creative it is when people operate properly at the top. It's a better way to work than power-playing, when people are slightly less talented and a lot more challenged."

I ask how his passion project is going: an adaptation of Flann O'Brien's At Swim Two Birds. It was due to shoot in 2011, but has hit more roadbumps than a milkfloat. Gleeson wants to direct from his own script, currently on its 14th draft; Colin Farrell, Michael Fassbender, Gabriel Byrne and Cillian Murphy have all been linked to star. You can see the disappointment it hasn't yet happened on his face; but when he speaks about it, it's all in terms of not letting someone else down, of accommodation and compromise.

It's this keen sense of responsibility that helps explain his keeping schtum about his own beliefs. His "duty of care" is to the movie; he doesn't want to abuse his position of power – or, at least, of publicity. "What I voice, I voice though my art, if that's not too vainglorious a word. But I don't think it is. That is my contribution. I don't think it's helpful for me to be a public spokesperson, to sound off without being part of the solution."

But he did break that vow once, in 2006, when he "went mad" on Ireland's The Late Late Show and railed against what he considered the poor state of healthcare in the country. The outburst was prompted by the last months of his mother-in-law, who was dying in a cancer ward when nurses pushed a coffin past the bottom of her bed. "It's like how not to do something."

He contrasts this with the "gift of a beautiful death" given to his own parents, both of whom died in hospices. "The boundless care the nuns gave my mother. She said she hadn't been creamed so much since she was a baby." He smiles, remembering her dry skin. "The horror of a death without dignity has so much implications for the people who are left behind.

"I hope I'm worthy in my dying. I hope I can maintain myself – that I wouldn't become pathetic and needy, and the worst part of myself come out in adversity. But I'm not afraid of it. It'd be such a silly thing to do! To ruin the life you have by fearing its ending."

Even the most personal of Gleeson's speeches can sound like a rallying cry. He stands and shakes my hand again, then sends me off into the hurricane, a general steeling his soldiers, a man to place your faith in.

• Calvary is out on Friday.

• This article was amended on 9 April 2014 because a reference to Brendan Gleeson's character in The Guard liking Russian literature was incorrectly introduced during the subediting process.