Speaking in the battleground state of Pennsylvania, Mr Biden said the US was facing multiple crises, which "under Donald Trump, keep multiplying".
The Democrat's comments came amid sharp tensions between the rival candidates over unrest in US cities.
Mr Trump has made "law and order" a major theme of his campaign.
The two candidates have in recent days been trading insults over clashes in Portland, Oregon.
On Monday, police identified the man who was fatally shot during Saturday's unrest as Aaron Danielson. His death was mourned by the far-right group Patriot Prayer, which described him as "a good friend and a supporter".
Police have not specified whether the shooting was directly linked to the clashes between a pro-Trump rally and Black Lives Matter protesters in the city.
However, media reports say a man who calls himself an anti-fascist is being investigated over Danielson's death.
Portland has become a flashpoint for demonstrations against police brutality and racism since the police killing of African-American man George Floyd in Minneapolis on 25 May triggered a wave of national and international outrage.
Mr Trump is set to visit the city of Kenosha, Wisconsin, on Tuesday amid anger there over the police shooting of black man Jacob Blake.
A teenager has been charged with killing two protesters during the unrest last week. His lawyers said he would fight the charges on the grounds of self-defence.
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In his speech in Pittsburgh on Monday, Mr Biden said the president "long ago forfeited any moral leadership in this country".
"He can't stop the violence - because for years he has fomented it," he said, adding: "We need justice in America. And we need safety in America. We are facing multiple crises - crises that, under Donald Trump, keep multiplying."
"Does anyone believe there will be less violence in America if Donald Trump is re-elected?" he asked.
Mr Trump has sought to project himself as the law and order candidate in November's election, and says it is the Democrats and Mr Biden who are soft on crime and on the violence that has sometimes flared at anti-racism protests.
Reacting to Monday's speech, he accused Mr Biden of "blaming the police" for protests and violence in the country.
"Just watched what Biden had to say," he wrote on twitter. "To me, he's blaming the Police far more than he's blaming the Rioters, Anarchists, Agitators, and Looters, which he could never blame or he would lose the Radical Left Bernie [Sanders] supports!"
In a tweet earlier on Monday, Mr Trump said "Radical Left Mayors & Governors of Cities where this crazy violence is taking place have lost control of their 'Movement'."
Mr Biden ridiculed Mr Trump for linking him to the violence, asking in his speech on Monday: "Do I look like a radical socialist with a soft spot for rioters? Really?"
He also condemned the violence that has sometimes accompanied the racial justice protests, saying he had personally spoken with the families of George Floyd and Jacob Blake who told him "none of this violence respects or honours" them.
The Democratic candidate called Mr Trump "an incumbent president who sows chaos rather than provides order" and a "toxic presence on our nation...poisoning our very democracy."
The Biden campaign's main line of attack has thus far been focused on the White House's handling of the coronavirus pandemic and on the fact that more than 180,000 people in the US have so far died with Covid-19. But Monday's speech is being seen as an attempt to fight Mr Trump's law and order message head-on by directly questioning his record on public safety.
Mr Biden's visit to Pennsylvania - an important swing state - marked an unusual trip for the Democratic candidate, who has worked mostly from near his home since coronavirus began spreading widely in March.
He has a stable lead over Mr Trump in the national polls. BBC
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US election