Diane Abbott, draped in a giant Brazilian flag, is in the hallway of a primary school and waiting to join a World Cup parade around the streets of her Hackney North and Stoke Newington constituency. "He thinks he's the dauphin, the young master," she says of David Miliband. She reveals that he insisted on having a meeting to establish what the duties of the new leader would be at the party conference in September (when the winner of the Labour leadership contest will be announced). "It was so he could organise his diary," Abbott says, rolling her eyes at her rival's assumption of victory.

Earlier, I had met Abbott at her house in Hackney. Her son, James - one of the most talked about of all MPs' children, thanks to his mother's decision to send him to a fee-paying school - opened the front door. Abbott is exhausted by talking about her son's education. "It's a subject that both the left and right bang on about because they seek to undermine the candidacy," she says.

She very publicly resisted offering an explanation for her choice of school for her son in an interview late last month with Andrew Neil for BBC1's This Week, the politics show that she usually appears on, sitting in alarming proximity to Michael Portillo. Neil challenged her earlier assertion that Caribbean mothers would go to the wall for their children and accused her of having been racist.

“I don't think you want to allow Andrew Neil to frame the debate around the Labour leadership," she says now. When I ask if they are friends again, she says they never were so in the first place. The unlikely trio of Portillo, Neil and Abbott were united, she says, by being products of grammar school - "outsiders".

In hustings and interviews since she announced her candidacy for the leadership race (on Radio 4's Today programme on 20 May), Abbott has made much of her difference and of her sense of being apart, both from the Westminster mainstream and from her fellow candidates. The pictures that adorn her bathroom walls tell their own story. They show her as a student at Cambridge University, in Labour Party line-ups and at political meetings. Abbott's is invariably the only black female face.

For many, including her fellow candidates (David Miliband helped her win the required number of nominations), it is what made her candidacy essential. But Abbott also emphasises her stark political differences. She describes the other four as being of "the political class" - all former policy advisers and part of the New Labour inner circle.

“They're all continuity candidates, but it's most striking with David Miliband because he has the same old Blairite funders, the same old Blairite money . . . That's a clue to where David is politically."

She continues: "The culture of briefing [inside the party] emerged because people are so close politically; it's the only way they can distinguish themselves." She feels "a million miles away" from such activity, and could not possibly say who was responsible. But the similarity of her leadership rivals distresses her, and she sees it as being a direct result of 13 years of New Labour. To be in cabinet under Tony Blair or Gordon Brown, you had to "have played the inside Westminster game". Neither leader was interested in promoting people from diverse backgrounds, with varied professional experiences.

If she became leader, she would recruit a shadow cabinet of "all the talents" and adopt Harriet Harman's idea of a 50 per cent female cabinet. She wants the party leadership to re-engage with the grass roots. "We have to revive democracy in the party; we have to allow the party once again to debate policy. For 13 years the party was kept at arm's length."

She criticises how this year's election manifesto was written by Ed Miliband with limited input from party activists. "Well, that's fine, Ed Miliband is very nice and very bright, but come on. We're meant to be a Labour movement."

Abbott is too easily dismissed on matters of policy by many of the insiders she resents. She is far to the left of her rivals: she voted against the Iraq war (unlike Andy Burnham), is in favour of scrapping Trident (unlike all the other candidates) and is dismayed by Labour's record on civil liberties. Differences on policy provoked her to stand for the leadership. "It was what Ed Balls was saying about immigration and the fact that other candidates were echoing it," she says. "That was the tipping point for me."

Then the matter arose at the New Statesman hustings on 9 June, and Abbott expressed disgust at the suggestion that Labour would have done better in the general election if candidates had been tougher on immigration on the doorstep, the applause was rapturous. And that is the unpredictable element about Abbott - there's an assumption that she won't win, that she's the rogue contestant, patronised into candidacy, but in fact her views often resonate strongly with the party faithful.

For her part, Abbott is optimistic. She says she has "not considered not winning", nor what her plans will be beyond September. Whatever happens, there is no doubt that she is much loved in her constituency, which re-elected her with an increased majority in May.

As we travelled to the school carnival on the bus, fellow passengers recognised her, shook her hand and wished her well. Her mantra is that "the party is only as strong as its base". What the other candidates might do well to remember is that the base likes Abbott a lot.

The questionnaire

If you weren't a politician, what would you be?
A novelist.

What's your favourite meal?
Chicken, rice and peas.

What does God mean to you?
I'm not a regular churchgoer, but I come from a family of regular churchgoers. I take questions of faith very seriously. However, I don't think you can ever use your faith to mask bigotry.

Have you ever taken drugs?
No.

What is your most valuable possession?
My house.

What would you consider your biggest fault or character flaw?
I tell jokes, and I find that people don't appreciate them.

Who will you advise your supporters to give their second preference to?
I haven't decided yet.