If you thought that Robert De Niro had mellowed in his old age, think again. His new film with Luc Besson is feistily violent and he still hankers after making a sequel to Taxi Driver

To misquote Bananarama, Robert De Niro is waiting inside the hotel room, talking on his phone, though probably not in Italian. I'm outside with the PR, who keeps easing open the door to check if he's done. The publicist is starstruck; he doesn't want to intrude. He explains that he grew up watching De Niro movies and that Taxi Driver is basically the reason he got into this business to begin with. We agree that it's wise to make no mention of this. He might shut the door and lock us out altogether.

De Niro is in town to discuss his new role as an ageing bull in a witness protection programme, and this seems fitting. Over the past four decades we have known him as sibilant Vito Corleone, volcanic Jake LaMotta, oily Rupert Pupkin, and any number of others, which is another way of saying we don't know him at all. De Niro hurls his work at the screen and then hides out in the shadows, unwilling (or unable) to talk the world through the process or police his own image. "With every other actor, I kind of know what they'd be like in real life," Billy Bob Thornton once remarked. "But with Bob De Niro I have absolutely no idea." He's the Moby Dick of American acting, forceful on screen and gauzy in public; a creature of splashy arrivals and murky descents.

At this stage it would be nice to hail De Niro's latest film as a late masterpiece, a movie to stir memories of the ones that came before. But this would only be half true. Specifically, The Family reminds us of Goodfellas to the extent that it installs Martin Scorsese's gangster classic as a jokey accompaniment to its own climactic scene. Yet De Niro is largely coasting as Giovanni Manzoni, a mafia supergrass who finds himself marooned in small-town France. The film is directed by Luc Besson, who leans heavily on the knockabout violence and accordion soundtrack. Guns are fired, baseball bats brandished and the rapacious Americans eat the locals alive. I stumbled out feeling as though I'd just been beaten about the head with a big string of onions.

Robert De Niro in The Family, directed by Luc Besson. Photograph: Jessica Forde

His phone call complete, De Niro duly motions me in, like a world-weary dentist faced with one last appointment. He is wearing silver-frame specs and a chocolate shirt and his once lithe frame has turned soft at the edges. At the age of 70, he knows the clock is ticking and that there's no time to waste. He suggests that the best thing about Besson is that he shoots his films fast. Shout action, call cut and put the scene in the can.

Experience has taught him that this is usually for the best. There is a danger of over-thinking a role; of getting hung up on details. "I used to worry about doing take after take," he says. "Now I don't worry about it so much. You can get it in the first take, more often than not. If there's some aspect of the character that you're really trying to get?" He shrugs. "Then maybe have two."

Three minutes in, I'm already wrong-footed. It's like hearing Pope Francis dispute the virgin birth, or learning that Vladimir Nabokov wrote with a set of kids' crayons. Wasn't it De Niro, more than anyone else, who took the art of immersive film acting to uncharted new depths? He learned phonetic Sicilian for his Oscar-winning role in The Godfather Part 2 and piled on 70lb to secure the best actor award for Raging Bull. He is renowned for his obsessive research and attention to detail; for disappearing into the role like some brooding Cheshire Cat.

I worry that, on some level, fame must have been the worst thing to happen to him. All of a sudden his face is too visible and his cover is blown. Maybe his success is both blessing and curse.

"Yeah, well, it can be a problem," he concedes. "In the beginning it was hard to deal with all the attention. As an actor it's hard to do certain things because you're well known. Research and stuff like that. That's the downside. The upside is that people are actually more forthcoming. When you ask them about the work that they do, they see it as you immortalising them. That makes them more honest."

Robert De Niro as Vito Corleone in The Godfather Part 2 Photograph: Everett Collection/Rex Feature

He compares what he does to being a painter, or writing a novel. "With painting or writing, you've got the finished product right there, with nothing in between. With movies it's different, because it's you in the movie. So there's a whole other social situation that you have to deal with."

De Niro, as it happens, is the son of writers and painters. His mother wrote erotica for Anais Nin and pulp fiction for True Crimes magazine. His father – Robert De Niro Sr – was a figurative expressionist and sometime poet, a member of a New York scene that included Mark Rothko, Willem and Elaine de Kooning and Larry Rivers, the reputed "godfather" of the pop-art movement. De Niro allows that he had a bohemian upbringing. I'm guessing that Jackson Pollock was constantly calling round, drunk out of his skull and merrily urinating into the De Niro family hearth.

"No, no, nothing like that," he says. "My father didn't hang out with Pollock. The only one that he was really friends with was Larry Rivers, although I did discover that he painted Elaine de Kooning, or maybe sketched her in charcoal. So yeah, he knew her too. But he was never a regular at the Cedar tavern. That was the big watering hole at the time."

Recently he has been working on a documentary about his dad, who died in 1993 without ever quite achieving mainstream success for his work. The actor says that he made the film primarily for himself and for his own children, as a kind of family history. In the meantime he has kept his father's studio exactly as he left it.

In making the documentary, did he spot similarities between his style of acting and his father's way of painting? "Yeah, there might be," he says. "There might be certain things. I mean, I didn't see my father paint very much. But I'd watch him sometimes and he was very intense when he did it. So there might be a connection with my father there. A way of zero-ing in on a problem and examining every piece of the situation before I make a choice of how I'm going to tackle it. But I'm not sure, because I never really had a discussion about how he painted, I wasn't really interested at the time." He shrugs. "That's what happens, kids aren't interested. It's a shame."

Robert De Niro as Travis Bickle in Taxi Driver: 'There was something about that guy.' Photograph: Ronald Grant Archive

How did his father feel about his own success? "Oh, he was happy for me," De Niro says. "He was proud. Maybe a part of him was a little envious, because he felt he was a great artist and …" He trails off. "His name was out there in the world," he says. "But it was my name as well."

By the time he turned 30, De Niro was already a star. He made a galvanic entrance in Martin Scorsese's Mean Streets, swaggering in slow-motion through the bar as Jumping Jack Flash wailed on the jukebox. It was the start of a fruitful collaboration. The director and actor would later re-team for Taxi Driver, a tour through the inferno of mid-70s Manhattan, brilliantly scripted by Paul Schrader. De Niro starred as lonely Travis Bickle, whip-thin, wild-eyed and thrumming with violence.

Today, I suspect, it remains the film that matters most, the one he'd most like to revisit. "Yeah, well, I had that idea," he explains. "I said, why don't we write something? And I talked to Marty and Paul did take a shot at something, whether it was an outline or a script, I forget. But somehow we didn't feel it was right and it didn't take off." He thinks it over. "But I'd like to see where Travis is today. There was something about the guy – all that rage and alienation, that's what the city can do to you. I mean Marty and I are from New York, and even we can feel alienated."

I'm fascinated to hear there could even be a sequel. It has been argued that Travis actually dies at the end of Taxi Driver and that the hazy closing sequence (the killer as hero, back behind the wheel) is just his fantasy of the afterlife. De Niro blinks behind his glasses. "You mean the end of the movie? Well, that's an interesting theory. I know that was not the intention, but it's as valid as anything." He's like a kindly professor correcting a student.

Robert De Niro in The King of Comedy Photograph: Allstar/Cinetext/20 CENTURY FOX

On starting out as an performer, he trained with Stella Adler, the legendary New York acting coach, who described him as the best student she ever taught. For some reason, the mention of Adler puts De Niro at his ease. He talks about her teaching methods, how she felt that the Method school was "too indulgent, too much cult of personality" and how the written word was sacrosanct. "She was totally about plays," he says. "The play is the play, and the playwright has final say, and you don't change a word because it's written in stone. The great plays have a message and the message is political. She had a class called 'script analysis' where she would boil a play down to its bare essentials and work towards building a character from that. I'd never experienced anything like it before."

It's interesting to hear him talk about theatre. I've always had the sense that he wasn't a fan. "No, I like movies," he says with a chuckle. "I mean, I'd do a play if I could find a great play, a modern play, a new play. But you can do more with film. I like the illusion. I like that you can create something and do it over and then put it together like a big puzzle. With a play, the most you can do is videotape it once and then put it in the archive at the Lincoln Center. Films last. You put it on a screen and it's there for ever, a little piece of history."

The appointment is over; he has places to be. De Niro hauls himself up from the couch, shakes hands warmly and says that it was good to meet me, he enjoyed our talk. He sounds sincere, although he may well be acting. That's the thing about De Niro. We don't when he's on and we don't know when he's off – and this, I suppose, is what makes him unique.

• The Family is released on 22 November.

• Robert De Niro on Silver Linings Playbook

• Quiz: how we do you know De Niro's films?

• De Niro's Tribeca mission