In a sometimes sombre address to the Conservative conference in Birmingham, Mr Cameron warned that the UK faces an "hour of reckoning" in which the decisions it makes will determine whether it will "sink or swim, do or decline".
The effects of the downturn, coupled with the rise of new economic powers around the globe, mean that Britons can no longer assume that their country will be able to continue to earn its living as a major industrial country, he said.
But he told delegates that he was confident that Britain will "rise to the challenge" and harness the "individual aspiration and effort" of its people to ensure prosperity in the future.
He promised to support "the doers, the risk takers, the young people who dream of their first pay-cheque, their first car, their first home and are ready and willing to work hard to get those things".
In deeply personal passages, Mr Cameron revealed how the example of his disabled father taught him the keys to success and to Britain's recovery - "Hard work. Strong families. Taking responsibility. Serving others."
Mr Cameron acknowledged that the Government's deficit reduction plan was "taking longer than we hoped", which he blamed largely on the eurozone crisis.
But he insisted that Britain was "on the right track", saying: "Yes it's worse than we thought, yes it's taking longer, but we are making progress."
And he insisted that the country will overcome its challenges, because "at our best, we're unbeatable".
Citing the summer of the Jubilee, Olympics and Paralympics, he told delegates: "Though the challenge before us is daunting, I have confidence in our country.
"Why? Because Britain can deliver. We can do big things."
And he promised to reform welfare, planning and education to unlock enterprise and growth and support aspiration.
"Let us here in this hall, here in this Government, together in this country make this pledge - let's build an aspiration nation. Let's get Britain on the rise.
"Deficit, paid down. Tough decisions, taken. Growth, fired up. Aspiration, backed all the way.
"We know what it takes to win - to win in the tough world of today, to win for all our people, to win for Britain. So let's get out there and do it."
Mr Cameron was scathing about Labour's Plan B to stimulate growth by borrowing more for investment, which he described as "a massive gamble with our economy and our future".
Dismissing Ed Miliband's bid to snatch the Tory One Nation slogan for his party, he denounced Labour as "the party of one notion - more borrowing".
And he rejected accusations that the Conservatives were the party of the rich and privileged.
While other parties "sneer" at those who want to get on in life, Tories "salute" them, he said.
"They call us the party of the better-off," said Mr Cameron. "No: we are the party of the `want to be better off', those who strive to make a better life for themselves and their families - and we should never, ever be ashamed of saying so."
As a party which had been led by grocer's daughter Margaret Thatcher, music hall performer's son John Major and the Jewish Benjamin Disraeli at a time of discrimination against Jews, the Conservatives have shown that "we don't look at the label on the tin; we look at what's in it", he said.
"We don't preach about one nation but practise class war, we just get behind people who want to get on in life."
After a conference in which the Tories have been accused of swinging to the right, with £10 billion in welfare cuts, new protections for householders who attack burglars and proposals for new restrictions on migration, Mr Cameron insisted he had not ditched his commitment to "compassionate" conservatism.
The Conservative Party was "for everyone, north or south, black or white, straight or gay", he said.
He defended his commitment to increase spending on international aid, while recognising the "scepticism" of many in his party about the pledge.
And he insisted: "Be in no doubt: this is the party of the NHS and that's the way it's going to stay."
He won applause from delegates by praising the role of the armed forces in Afghanistan, the Queen's involvement in the Jubilee celebrations and her parachuting cameo in the Olympics.
And he even had warm words for Boris Johnson, who has been touted this week as a rival for his job, hailing him as "the man who put a smile on our face, the zinger on the zip-wire".
But the bulk of the speech was devoted to a frank analysis of the economic situation and the action he believes Britain needs to take to secure success in the future.
The coalition Government came to power at "a grave moment in the modern history of Britain", faced with a challenge to restore national solvency, bring troops home from war and mend a broken society, Mr Cameron said.
"Two and a half years later, of course I can't tell you that all is well, but I can say this: Britain is on the right track," he said.
He warned: "All of my adult life, whatever the difficulties, the British people have at least been confident about one thing. We have thought we can pay our way. That we can earn our living as a major industrial country and we will always remain one.
"It has fallen to us to say - we cannot assume that any longer.
"Unless we act, unless we take difficult, painful decisions, unless we show determination and imagination, Britain may not be in the future what it has been in the past.
"Because the truth is this. We are in a global race today. And that means an hour of reckoning for countries like ours.
"Sink or swim. Do or decline."
He said that he wanted to set out a "serious argument" explaining "how we compete and thrive in this world, how we can make sure in this century, like the ones before, Britain is on the rise."
"Nothing matters more," said Mr Cameron. "Every battle we fight, every plan we make, every decision we take is to achieve that end - Britain on the rise."
