Joseph Beuys, How to Explain Pictures to a Dead Hare (1965) © Marina Abramović

Meeting anyone famous is nerve-racking. Greeting someone who has got naked, cut, burnt, stabbed, obsessively scrubbed bones and in one case almost died when a performance went wrong, is intimidating. So it is with some trepidation that I go to meet the performance artist Marina Abramović at her office/studio in downtown New York. The last time we met was in 2011, shortly after her triumphant show which attracted more than 850,000 visitors to the Museum of Modern Art in New York.

I ask her whether she still has her loft nearby and a quick cloud passes over her pale face. "No, that is all gone; I got divorced [from the Italian artist Paolo Canevari] and I have sold – given everything away". Her offices in Soho are populated by her team, clad, like her, all in black, with the exception of her IT man who dresses daringly in a checked shirt. The team's mission is to help to sustain the Marina Abramović "brand" and to create and build a performance centre, the Marina Abramović Institute (MAI), in Hudson, New York.

Today, however, we are here to talk about plans for her forthcoming show at the Serpentine Gallery in London. Those who go to the gallery this summer will encounter an empty space – empty except for the performance artist. For more than two years, Abramović has been in discussion with Julia Peyton-Jones, co-director of the Serpentine, and Hans-Ulrich Obrist, co-director of exhibitions and programmes and director of international projects, about the content of the show. They decided in the end to build on the work The Artist is Present, previously presented at MoMA, in which Abramović sat gazing into the eyes of visitors to the exhibition. "When you come and sit on the chair everything dissolves. You don't see any people, you don't hear the space. You see your face, then you see my eyes and then my eyes disappear. You become all but alone. I am just triggering all emotions in you. It becomes like a mirror."

At the Serpentine even the two chairs and table used at MoMA will be absent. "No timetable, no lab coats. I'm dressed the same as everybody else because I don't want to say 'the woman in the white coat, she is the artist'. I want only the difference between them to be this energy I am trying to project, which could be, you know, different."

No one was prepared for the furore caused by the New York show; those who queued and sat more than 10 times formed a club and still meet regularly to recount their experiences. The celebrities who attended included Isabella Rossellini, Björk and her then partner Matthew Barney, and Lady Gaga, and all obeyed her rules of silent engagement. The power of the gaze was apparent though. Ulay, Abramović's collaborator from the 1970s and 1980s, reunited briefly with her at the opening of the MoMA show, sitting opposite her. She was unable to hold her pose, grasping his hands, visibly moved by his presence.

In London, Abramović will personally open the gallery at 10am and close it at 6pm. As the visitor enters there will be lockers to leave personal belongings, including watches, phones, laptops – all the accoutrements of modern life that mark time. The Serpentine will be empty, but the park will be full of eager participants waiting to enter; how to entertain the waiting is something that is already being considered by the Serpentine team.

Today, however, Abramović herself is engaged with calls and texts on her mobile phone as her assistants remind her on a regular basis of time passing.

She is in training. It is a "fast day", so she is sipping water and eating pills, following a programme designed by Nasa. "I need to be in shape for this event as it all rests on my energy." Before the show opens in London she is returning to Brazil to complete her study of shamanism, The Current. "I go to Brazil to look for places of power; for the people. The people who have certain energy; the ones who really learn how to take energy from outside and through inside of themselves, transform it and give it back to the ones who don't know how to do that. I'm learning how to move the wind."

Marina Abramović was born in Belgrade, in the former Yugoslavia in 1946. "When I was born my mother, because she was making her political career, just gave me to my grandmother to take care of me. I didn't see my mother and father until I was six." Her parents were both staunch Communists and her grandmother was obsessively religious: "She was always going to the church. I spent my childhood in a church following my grandmother's rituals – candles in the morning, the priest coming for different occasions."

When her brother was born, when Abramović was six, her parents brought her home. "My mother and father were national heroes so we had this privilege. I had the piano lessons, the French lessons, the English lessons, all the private things." That she did not have art lessons did not matter, "as the only thing I wanted to do was to be an artist. There has never been any doubt in my life that I could be anything else".

At home there were chores to do and strict rules to be adhered to; her mother left her a list on top of the piano every day. "I have to learn 10 sentences of French. I have to wash my hands so many times a day. So everything was instruction." She had a curfew of 10pm but managed to do her performances and still be home, until eventually she fled aged 29. When she was reported missing to the police, their response was laughter: "Your daughter has escaped!"

Abramović arrived in 1970s Amsterdam and "there were hippies and all sorts of things. So I got very lost in the beginning because restrictions didn't mean anything there. All my work and my ideas about art, about rebelling didn't work there". But what she did learn was the power of self will. "That you can do anything if you set your mind to it. That is something that I apply now to performance art."

In Amsterdam, in 1976, she met Uwe Laysiepen, known as Ulay, who became both her lover and collaborator. "Firstly, it is a big love story. We were born on the same day, and we met on our birthday." Their works explored both the power of their individual gender and their sexuality. Works that linger in the memory include Imponderabilia, where they stood naked facing each other in a doorway so that visitors had to choose who to face while squeezing through the slightly-too-small entry point.

"Basically I realised two people can do more than one," she says now. They stayed together and worked together for 12 years, but the closeness put a strain on their relationship, so in the end a work, sadly and rather ironically entitled The Lovers: the Great Wall Walk (1988), that was initially meant to act as a marriage ritual, became instead a marking of their separation. "The Great Wall Walk is a product of this separation because it was the product of the end of our relationship". When I ask how painful this was – a walk of 2,500km taking 90 days along the Great Wall of China, followed by a painful embrace and separation – her response is heartfelt. "Oh God! I was 40. I wrote in my diary. I was 40. I was fat, ugly and unwanted."

It was out of the pain of this separation that the idea for The Life and Death of Marina Abramović, a theatre piece with Robert Wilson, evolved. It was, she says, "the only way to deal with this after being so depressed. I realised I wanted to make a theatre piece".

This work brought her biography to life; her strange upbringing and the obsessive religion were described by a narrator, Willem Dafoe, accompanied by original songs by Antony Hegarty of Antony and the Johnsons, with Abramović playing herself in the main role.

With a musical dedicated to her life, the success of her show at MoMA, and her appearance on Time magazine's list of the 100 most influential people of 2014, one of only two artists (the other being Steve McQueen), it is clear that performance, or at least performance by Marina Abramović, has become a mainstream activity, and one that is increasingly gaining a place in the centre of the discourse of contemporary museums.

Abramović is clear about her own importance, and she is also determined to leave a lasting legacy. "I am like a brand, like Coca-Cola, the Fulbright Foundation, Lincoln Centre, whatever you want. So it is not about sculpture, but performance." One of the hurdles for performance art is its ephemeral quality. How you archive and retain the evidence of past performances is an issue that the MAI – the Marina Abramović Institute or the "Many Artists Institute" – plans to address. She has found a building in Hudson, a small town two hours from New York City, and has hired the architect and theorist Rem Koolhaas to renovate it. Within the institute will be a workshop dedicated to teaching the Marina Abramović method. "I found out that this is the great space for creating this... it's not a school, even. I will call this place a cultural spa."

Using Kickstarter, the MAI has raised $660,000 (£391,000), more than doubling the previous record of funds raised in one month by the fundraising platform, and making the front page of the New York Times – something that Abramović is obviously proud of, as she digs out the newspaper to show me. She received criticism for using Kickstarter in this way, but she quickly dismisses it.

"Anything that will get the message of the importance of performance is OK," she says earnestly, writing down for me the link to the BBC interview in which Lady Gaga talked about Abramović's importance as a mentor to her. "They don't understand that I always have a big picture of why I am doing certain things. Look at Lady Gaga, I mean, Meat Dress. You know there are at least six artists from the Seventies who made meat dresses? But they will never mention that. They will just mention her..." It is not surprising, therefore, to see Lady Gaga featured in a video describing the MA method, or to hear that Abramović is helping Lady Gaga on a future video.

Abramović is unrepentant about hanging out with Lady Gaga, James Franco (who calls her a deity) and Jay-Z, among other popular cultural figures. Later, I see a toe-curling video clip filmed at the Pace Gallery in New York in which Jay-Z pulls Abramović on to a platform to rap and dance with him. Abramović, who has made her career out of being the strong body at the centre of the performance, is here a fish out of water, flapping around like an auntie dancing at her nephew's wedding with his hipster friend.

Her institute will be both an archive for past performances and a conduit for future collaborations across science, literature, dance and theatre. Unlike many other performance artists, Abramović is a keen proponent of the restaging of works, both her own and others'. In her exhibition at the Guggenheim, she restaged the work of other performance artists, including Bruce Nauman and Vito Acconci. She says: "First of all, the nature of performance art is life. It is a living art form. If you don't re-perform that, it becomes a photograph in a book or a bad video."

Her ambitions are large: "I wanted to kind of figure out, if all these people pass through this kind of method, you know, and if I could change the way they think about their life, even in the smallest way, and how I can actually change the way of consciousness somehow, I could create a model society."

To this end, she has devised the Marina Abramović method, a series of exercises in endurance-training for long-durational works. One of her assistants shows me a presentation of the MA method, including an exercise separating rice from beans that you could do on a computer. Another is simply staring at an abstract painting for an hour.

When I query whether the Marina Abramović method that will be taught within the centre will be more prescriptive than the rules she had to obey growing up, she ducks the question, eventually saying: "But I just want to say, I left my rules in Belgrade. I went to these other cultures and I saw the rules were more strict than my own rules."

During the PowerPoint presentation shown to me by her director of media – the Marina Abramović method rolled out in a professional yet emotionless way – I thought of the performances, like Rhythm O in the early 1970s, where she gave control to the viewer, allowing them to choose what they did to her body with 72 instruments including a rose, a pen, a scalpel, honey, a feather, a gun and a bullet. Abramović has survived her continual assaults on her body and now has arrived where she cheerfully calls herself a "Coca-Cola brand", or the "grandmother of performance art". But I console myself with her words that "art has this incredible immateriality and what really moves you is not the money. It moves you with something else. It moves you with something that lifts your spirit".

Abramović is clear that to fulfil her art, self-discipline is key. "I understood that in these rules and repetition there's incredible power, because this is the way that we can actually control our physical body and free our mind and spirit."

But it is in the co-option of other artists' performances and their self- discipline that I become nervous. I watch the MAI presentation in which homage is paid to Tehching Hsieh, a Taiwanese performance artist. His performances, five in all, consisted of sustained long-durational activities, each lasting a year. Living outside, living in a cage, tied to another artist (Linda Montano) – after the five were achieved, he declared, "I have finished my art. I will live life now".

When I mention Hsieh to Abramović she says, "I have seen him recently. You must go and visit him – he now lives in Brooklyn. He made me little dumplings and then he showed me the huge boiler to heat the apartment building – he said 'I built this myself'." Talk about real heroic acts – but after time with Abramović, you feel that you, too, could build a boiler single-handed.

'Marina Abramović: 512 Hours' is at the Serpentine Gallery, London W2, 11 June to 25 August

The art of Abramović: three key moments

Rhythm 10, 1973

In her first performance piece, Abramović played a game in which she aimed stabs of a knife between her splayed fingers. Whenever she cut herself, she would continue with another knife until she had used 20. She would then play back a tape recording and repeat the ritual.

The Artist is Present, 2010

A 736-hour and 30-minute piece in which Abramović sat in MoMA's atrium while spectators – including the artist's ex-partner and collaborator Uwe Laysiepen – took turns sitting opposite her for as long as they chose.

Picasso Baby, 2013

Last year, Abramović featured in Jay-Z's performance art film. It was, according to one commentator, "the day performance art died".