* Interior minister May bids for Cameron's job

* Ex-London major Boris Johnson expected to stand too

* May says politics is not "a game"

* Opposition Labour Party also facing leadership contest

By Kylie MacLellan

LONDON, June 30 (Reuters) - Britain's interior minister Theresa May put herself forward on Thursday to replace Prime Minister David Cameron, promising to deal with divisions in the country that drove more than 17 million Britons to vote to leave the European Union.

Cameron, who led the Remain campaign, announced his resignation after Britain voted by 52-48 percent to leave the EU, triggering a leadership contest within the ruling Conservative Party that will elect his successor by early September.

May, who is expected to face leading Leave campaigner Boris Johnson in the battle for the top job, made a thinly coded attack on the ex-London mayor's privileged background and pointedly said government was not "a game".

"If you're from an ordinary, working-class family, life is just much harder than many people in politics realise. You have a job, but you don't always have job security," said May, who has impressed many Britons with her handling of security issues as interior minister for the last six years.

"Frankly, not everybody in Westminster understands what it's like to live like this. And some need to be told that what the government does isn't a game, it's a serious business," she wrote in The Times newspaper.

The Brexit vote has triggered one of Britain's biggest crises in modern times with political upheaval, more than $3 trillion wiped off global stocks and one of the steepest falls in sterling in a generation.

Both major political parties are in turmoil after the vote, with Labour Party leader Jeremy Corbyn likely to face a direct challenge for his job from colleagues after losing a vote of no-confidence in his leadership.

Recognising the divisions in Britain, Conservative May said she would make Britain work for everyone and promised big changes, if she won the top job, for the economy and society.

"We believe in capitalism and free markets, for example, because history has shown them to be the best way in which we spread opportunity and improve social mobility," said May, the state-school educated daughter of a Church of England clergyman.

"But where capitalism is not helping to provide opportunity for all, where it is losing public support, where there are gross abuses of power, we need to reform it."

May supported Cameron's Remain position but was not one of the main campaigners for staying in the EU.

Both Cameron and Johnson were educated at Britain's most exclusive school, Eton College, and knew each other at Oxford University. May read geography at Oxford.