When Pablo Trapero was 13, growing up in the rough-edged district of La Matanza outside Buenos Aires, his father gave him a warning. It was 1985 and Argentina had just emerged from the cruel years of the military junta. The boy wanted to get involved in his school council. His father, sceptical of how long democracy would last, advised against it. If the generals came back, he said, an interest in politics could end badly.

That summer, Trapero watched a strange story unfold on the TV news. Several members of a wealthy family from the capital had been arrested for the kidnapping of four people. Three had been murdered despite ransoms being paid. “I was young,” he says, “so I don’t think I really understood it. But it stuck in my mind.”

Now Trapero is a successful director, the maker of acclaimed movies about the dark corners of story. The family were the Puccios, a name still notorious in Argentina. The father, Arquímedes, proved to have planned the crimes – but he didn’t act alone.

The first victim was Ricardo Manoukian, a friend of the eldest son, Alejandro Puccio. The pair played rugby together; Alejandro had been good enough to be picked for the Argentinian national team. The family imprisoned Manoukian at their gated home, accepted a ransom of $250,000, then shot him. The same method was used twice more, until the police finally intervened while a ransom was being paid for the fourth victim. Soon, it transpired that, in the time of the junta, Arquímedes Puccio had worked for the government in their “dirty war” of terror and disappearance against tens of thousands of trade unionists, students and activists.

Trapero is bearded, shaggy and, for all the horrors of his film, warm and affable. The surviving members of the Puccio family, he says, did not help him. In fact, they denied being Puccios at all. “The mother and one daughter still own the same house, but they changed their names. Our production team contacted them and they literally said, ‘Who? No, you have the wrong address.’ And closed the door.” But Trapero had other sources: social acquaintances, team-mates of Alejandro, lawyers offering case files and recordings of the calls made to the victims’ families with ransom demands. Those families were also involved in the film, visiting the set to confirm details of location or costume. Trapero says that process was creatively thrilling. “But then you remember that for them, reliving this is the most terrible thing.”

Without a detective to build his story around – the police were scarcely involved until the end – The Clan instead takes us inside the Puccio home, a display case of middle-class banality. With the comic actor Guillermo Francella deftly cast against type, Arquímedes is a shark-eyed torturer and killer, but around the dinner table just a pompous old man. The other Puccios affect not to notice the screams from the top of the house. Alejandro complains bitterly about his father involving him in murder, then uses the proceeds to open a sports shop.

Making the film reminded Trapero of his father’s warnings. The dirty war is assumed to have been a busy, bloody time for Arquímedes Puccio. Then, as the country became a democracy, he simply slipped into the role of affluent businessman, still preying on his fellow Argentinians, but now for private profit. What Trapero calls “the people in the shadows” remained obscenely powerful. “I am more afraid of this kind of person than I am of the bastard who attacks you openly,” he says. “The person whose face you don’t even know.”

For Trapero, there was always more to the story than one lone sociopath. When the tapes of the ransom demands surfaced, he says, it turned out they hadn’t been recorded by the police, the Puccios or the victims’ families. The list of other candidates was short. “Those who Arquímedes worked with in the junta, they were always part of these crimes. And they took their share of the money.” So there were people involved who were never arrested? “Absolutely. Completely.”

Before the trial of Arquímedes, Alejandro and another son, Daniel – the remaining family members were never charged – sections of the Argentinian press scoffed at the idea this perfect family could be guilty. Thanks to the Puccios’ legal team, Trapero says, the trial itself “was really a negotiation”. After his conviction, Arquímedes enjoyed a prison life deeply relaxed by Argentinian standards, in which he successfully completed a law degree. “There were always these favours.” So while Trapero was “terrified” about the response of the victims’ families to the film, he says they were delighted. “I feel uncomfortable saying it, but one of them told me, ‘Thank you – this is real justice.’”

When Trapero was a child, his mother taught and his father sold cars. In English, La Matanza means “the killing”, a reference to it having once been a place where cattle were slaughtered. Under the junta, it also hosted pockets of concealed opposition. The church school Trapero attended was influenced by leftwing “third world Catholicism”. Only years later did he learn that some of the priests who taught him had been sheltered by the school from the military. The experience inspired the film he made before The Clan, White Elephant, about a shantytown priest played by the great Ricardo Darín. The same actor also starred in the movie before that, Carancho, the story of an ambulance-chasing lawyer, until now its director’s best-known work.

Despite the gravity of its story, The Clan is made with pizzazz: it has the heft of history and the zing of Scorsese. In Argentina, the reward was a box-office sensation. On its opening weekend, 53% of Argentinians going to the cinema went to see The Clan.

Trapero was surprised such a huge audience wanted to revisit so bad a memory. “But my theory about Argentina is that it has this postcard image of itself, which is very melancholy. We’re forever thinking about our broken dreams.” He smiles. “The importance of the tango explains that. It’s not like samba in Brazil, which expresses happiness. Tango is about sadness and obsession with the past. And in Argentina there is the highest ratio of psychologists to ordinary people anywhere. Everyone is in therapy. My wife studied psychology, my sister is a psychologist …” And you make films, I say. “Exactly! So I need therapy, too. But it means there is always reflection.”

He says he was glad to make the film for the sake of his son. Otherwise, “he might have one day heard this story and thought it was just fiction.” But for now at least, he wants to step away from Argentinian history. He says he feels optimistic about the country. “For my father’s generation, there was always the despair that the junta would come back. I don’t think that will ever happen. Maybe I’m naive. But actually, I prefer to be naive.”

• The Clan is in cinemas on 16 September.