Rothko revealed: Christopher Rothko shares troubled memories of his father Mark

The tapas restaurant that Christopher Rothko has chosen for our interview is on New York's Upper West Side, with tables that spill on to the street. It is very good, and much fancier than anywhere his late father Mark – the celebrated avant-garde painter whose abstract expressionism changed the face of art – would have dined for most of his life.

"People imagine my father had a glamorous existence, but he lived mainly in slums," Christopher says, as he settles into his chair. Mark Rothko's best-known paintings now sell for tens of millions of pounds, but during the Great Depression he lived hand-to-mouth, and until the last two decades of his life – which he ended with a razor blade, in his studio, early one morning in 1970 when Christopher was six – was largely unrecognised and unsupported by the art establishment.

The artist's 45-year-old son is a psychologist, but has for the past eight years looked after the Rothko Family Collection and Archive with his older sister, Kate. Rothko's prolific body of work, mostly produced before people began to think of him as an artist of note, was often done at night, after a day of teaching, and at weekends. Part of the tragedy of his career was that he spent a lifetime struggling for recognition, then struggled to deal with it when he achieved it. "I don't think he ever fully came to grips with it," Christopher smiles. "As soon as he became successful he began to suspect it, that maybe the painting was too easy because so many people liked it."

An episode in his career that epitomises Rothko's ambivalence about success is the Seagram murals. In 1958, Rothko was asked to supply wall paintings for the Four Seasons restaurant in its opulent new building in New York. He agreed. Yet, by 1960, he returned the portion of the $35,000 fee that had been paid to him, and kept the murals. Later, the artist is reported to have said that he accepted the assignment with "malicious intent", and hoped to paint something that would "ruin the appetite of every son of a bitch who ever eats in that room". (Four of the 40 he completed for the commission have hung in the Tate's galleries since he donated them in 1965, and now form the centrepieces of Tate Modern's forthcoming exhibition.)

Rothko, it seems, was uncomfortable with the restaurant's ostentation. "I think he deceived himself about what that restaurant was going to be about," Christopher says. "I think he desperately wanted to do a major public work ... but once he saw the reality that had always been whispering to him, he couldn't ignore it any more."

By the early 1960s, Rothko's work was being collected by the likes of the Rockefellers; yet, despite his high profile, he was all too aware of a shift in tastes and what it might mean for his artistic standing. (He is said to have called the Pop Artists "charlatans and young opportunists".) Indeed, his last significant project, a specially commissioned chapel to house specific work of his, was erected in Houston, Texas – a long way from the art world of New York. In the few years before his suicide in 1970, he even temporarily stopped working on his trademark large canvases – a near fatal aneurysm led his doctor to advise him to work on a less physically arduous scale.

What motivated Rothko through such a turbulent career? Certainly nothing as straightforward as organised religion. "Religion is something I've come to in my own way, but not through him," Christopher says. "My father let go of the institutional trappings of religion, but I think for him the artist had a quest for finding meaning: that's one of the things religion does, in a very basic kind of way."

Born Jewish, Rothko rejected organised religion in his adult life, yet was attracted to the transcendent. Not a voracious reader, he nevertheless repeatedly delved into Nietzsche's The Birth of Tragedy and Kierkegaard's Fear and Trembling, copies of which he kept in his studio. "I think that for him, painting was a means to a philosophical end," Christopher says. "Myth is basically how we explain the world around us."

Four decades since his death, interest in what drove Rothko, and in particular the moment he made the great leap to the art that brought him fame, is as keen as ever – not least because Rothko's stock has risen to the point where his 1950 work White Centre (Yellow, Pink and Lavender on Rose) sold last year for $72.8m. Along with his astonishing value to the art market have come the scholars, desperate to discover more about one of the key figures of abstract expressionism (a term the artist himself always rejected). Their attentions prompted Christopher finally to examine properly his father's written manuscripts. He eventually edited and published the writings, found only after Rothko's death, in 2004. (Christopher's first experience of his late father's estate was not pleasant. In one of the biggest art scandals of the 1970s, he and his sister successfully brought a law suit against the executors of the Rothko estate for its mishandling, resulting in a $9.2m fine.)

The writings span a period from the late 1930s to early 1940s; Rothko divorced his first wife, Edith, a jeweller, in 1943, marrying his second, Mell (the mother of Christopher and Kate) two years later. He was also about to leave behind the figurative art he had laboured at for years. As Christopher and Kate ploughed through the chaotic documents, they realised their value. "It [reads] like somebody who has been painting abstract painting for 15 years – not someone who has not quite made the break into abstraction," Christopher explains. "In his thinking about art he's already 15 or 20 years down the road."

Indeed, it was around this time that Rothko set aside the writing and started painting again after a year's break, producing canvases of blurred fields of colour layered one on the other, either horizontally or vertically: the "multi-form" paintings that were to develop into his spectacular signature style.

Trying to explain the force of his father's work over a huge plate of vegetable paella is a little incongruous, but Christopher manages it. "When it's really working well, you can go into a room and lose yourself standing there for 15 minutes before you realise you've been there more than two," he says. "Otherwise they're just rectangles, right?"

The Menil chapel in Houston – now known as the Rothko chapel – with Rothko's Black-Form paintings, is the place to go for the ultimate Rothko experience, he says. Christopher didn't go until 1996, and was unprepared for how intense it would be. "I stood there for three or four minutes and had this uncanny urge to run out of there," he says. "I've come to really love it since then; I won't say it's easy, but it's a familiar place now."

In a poignant thought noted down in 1947, Rothko considered that a painting "lives by companionship, expanding and quickening in the eyes of the sensitive observer. It dies by the same token. It is therefore a risky and unfeeling act to send it out into the world."

If he saw each painting as a something of an orphan, it is easy to draw parallels to his own experience: taken from Russia as a boy to a small town in Oregon, his father soon died. Christopher doesn't need to try hard to empathise: the loss of his own father was followed one year later by the death of his mother and a move from his home in New York City to the care of relatives in Ohio.

Aside from piecing together the man as an artist, Christopher's memories of his father are fond, and vivid. "He really loved music as much – if not more – than he loved art," he says. "I'd really taken on that passion and even from the age of two I'd be demanding that he changed the record – I wanted Schubert not Beethoven. We would have endless debates because he loved the Magic Flute beyond all operas and I always insisted that Don Giovanni was better. I wish he was here, not least because I want to continue that argument." He stops, and gives a small laugh. "I still think Don Giovanni is better."

'The Artist's Reality', edited by Christopher Rothko, is published by Yale University Press. Rothko, an exhibition at Tate Modern, Bankside, London SE1, runs from Friday (www.tate.org.uk). Clare Dwyer Hogg flew to New York courtesy of American Airlines

Art in the Apple: Rothko's New York

Mark Rothko moved to New York in 1923 and lived there the rest of his life. He taught in the Jewish Center School in Brooklyn for over 20 years, as well as Brooklyn College and Hunter College in Manhattan. His pupils, now in their seventies and eighties, remember him fondly, but say he didn't teach as much as tell them to paint what they liked.

For his own continuing artistic education, Rothko went time and again to look at the Pompeian wall paintings at the Metropolitan Museum of Art (above): these ancient pieces of art, in deep reds, shadowy blacks and greens, employ colours he would himself use.

You can't visit his studios as they were, but the tea-room in the Urasenke Chanoyu Center (East 69th Street) used to be one of Rothko's main places of work.

To get away from it all, the painter would often sit in the garden of the Museum of Modern Art (West 53rd Street) – he lived only a couple of blocks away at one point, and now his 1944 masterpiece Slow Swirl at the Edge of the Sea, painted for his wife Mell, is in the permanent collection. CDH