He said he had spoken to Karzai, whose re-election was confirmed this week, several times in recent days and urged progress on the key issues of security, governance, reconciliation, economic development and regional relations.
Brown said: "If the government fails to meet these five tests, it will not only have failed its own people, it will have forfeited its right to international support."
The prime minister said he had proposed that Karzai draw up a new anti-corruption law, with a new commission advised by a high-profile international figure to investigate fraud and set up new rules for the transparent awarding of contracts.
James Bays, Al Jazeera's correspondent in Kabul, the Afghan capital, said: "This follows other similar pressure on Hamid Karzai, we've heard in the last few days from President Obama, from the UN special representative [for Afghanistan] Kai Eide.
"All of them passing on the same message that the new Afghan government cannot continue business as usual, and particularly focusing on that issue of corruption.
"I think it will be interesting how the Afghan government responds to one point ... in the speech ... which is the suggestion that there should be a new anti-corruption agency.
"And he [Brown] mentioned the possibility of a figure of international stature coming in here to run that agency, now I think that will ruffle a few feathers."
Last month, a UN-backed election watchdog found widespread fraud had taken place during the country's August 20 presidential election.
A date for a presidential election runoff was announced only to be cancelled after Abdullah Abdullah, Karzai's political rival, withdrew his candidacy citing the government's refusal to reform the election commission.
|There is growing public unease in the UK about the Afghan mission [EPA]
Warning Karzai on government appointments, Brown said: "Cronies and warlords should have no place in the future of a democratic Afghanistan."
Brown's remarks came in the course of a bloody week for British troops in Afghanistan with seven soldiers killed since last Saturday.
Five were shot by an Afghan policeman on Tuesday, bringing to 230 the number killed since the US-led invasion in October 2001.
The prime minister repeated that the mission was to stop al-Qaeda from using Afghanistan as a base for attacks on Britain and elsewhere, and said it "must not fail".
"We cannot, must not and will not walk away," he said.
He said more Afghan troops were needed, adding: "The first priority of any government is to provide security for its people.
"It is not sustainable to subcontract that task indefinitely to the international community.
"So the expansion and training of the Afghan army and police must be the new government's first priority."
About 500 British troops are involved in mentoring Afghan soldiers and policemen to help them take over security from international forces, but this role was thrown into question by Tuesday's shooting.