WASHINGTON — President Trump unleashed a scorched-earth campaign on Thursday against former Vice President Joseph R. Biden Jr. hours before Mr. Biden was to accept the Democratic presidential nomination, predicting “mayhem” if his rival won the general election in November.

“At stake in this election is the survival of our nation, it’s true,” Mr. Trump told a small crowd in a Pennsylvania town not far from where Mr. Biden was born. “Because we’re dealing with crazy people on the other side. They’ve gone totally stone-cold crazy.”

The president took aim at Senator Kamala Harris of California, the Democratic nominee for vice president, seeking to stoke fear of immigrants by offering a brutal description of a case involving an undocumented immigrant who was included in a jobs program while Ms. Harris was the top prosecutor in San Francisco.

“As district attorney of San Francisco, Kamala put a drug-dealing illegal alien into a jobs program instead of into prison,” Mr. Trump said. “Four months later, the illegal alien robbed a 29-year-old woman, mowed her down with an S.U.V., fracturing her skull and ruining her life.”

It was a reference to the case of Alexander Izaguirre, an undocumented immigrant convicted of selling cocaine in 2008. He was a participant in Back on Track, an initiative spearheaded by Ms. Harris that placed young, first-time drug offenders who pleaded guilty to their crimes into a jobs program rather than jail.

Mr. Trump’s attack on Ms. Harris was an echo of the Willie Horton episode during the 1988 presidential race between Vice President George Bush and Gov. Michael S. Dukakis of Massachusetts. Mr. Bush’s campaign and its allies repeatedly invoked Mr. Horton — a Black prisoner in Massachusetts who, while released on a furlough program, raped a white woman — to suggest Mr. Dukakis was soft on crime.

In July 2008, Mr. Izaguirre snatched a woman’s purse and jumped into a car. When the woman jumped onto the car’s hood, the driver slammed on the brakes and the woman was thrown to the sidewalk, The Los Angeles Times reported in 2009.

When she learned of the case, Ms. Harris said that year that Mr. Izaguirre’s inclusion in the program was a mistake, one that she had since fixed by requiring would-be participants to provide “everything from Social Security cards to whatever it is that they can produce” to show that they are in the United States legally.

At the time, 113 offenders had successfully completed the program, while 99 had failed and been sent back into the court system. Ms. Harris said in 2009 that Mr. Izaguirre was one of seven undocumented immigrants in the program, but was the only one who failed, according to The San Francisco Chronicle.

As the president prepares for the Republican National Convention next week, he has begun to reveal the core of his coming campaign: a highly personal assault on Mr. Biden and his running mate.

“These people have gone insane and they are radical left,” Mr. Trump said, calling Mr. Biden “a puppet of the radical-left movement that seeks to destroy the American way of life.”

In Mr. Trump’s often blatantly misleading telling, Mr. Biden and Ms. Harris will destroy the country by taking guns away from Americans, funding late-term abortions, encouraging “deadly ‘sanctuary cities’” and allowing low-income housing to “abolish suburbs.”

Seeking to undermine Mr. Biden’s appeal in rural, blue-collar parts of Pennsylvania, the president lashed out at him, saying that Mr. Biden had “abandoned” his hometown, Scranton, when his family moved when he was a young boy.

Mr. Trump said that Mr. Biden had been at the forefront of a “globalist attack on Pennsylvania voters,” citing the former vice president’s support of trade deals with Mexico, China, South Korea and “the horrible, ridiculous Paris climate accords.”

“As far as I’m concerned,” Mr. Trump asserted, “Joe Biden is no friend of Pennsylvania.”

In his remarks, the president also ascribed to Mr. Biden’s many positions that he does not actually hold or he mischaracterized Mr. Biden’s proposals.

Mr. Trump misleadingly claimed that Mr. Biden had “pledged to hike your taxes by $4 trillion dollars.” Three-quarters of the proposed increases would be levied on the wealthiest 1 percent of the population.

President Trump falsely said that Mr. Biden would “take away your guns.” Mr. Biden’s proposed gun control measures include a buyback program that would allow those who own assault weapons and high-capacity magazines to either sell them to the government or register them.

Mr. Trump also falsely said that Mr. Biden would “give free health care to illegal aliens.” Mr. Biden’s health care plan would allow undocumented immigrants to buy government-sponsored insurance or private plans on government-run exchanges, but it would not be “free.”

President Trump misleadingly claimed that Mr. Biden would “close down charter schools.” Mr. Biden has said that he would end voucher programs for private, for-profit charter schools, but that he supports high-performing public charter schools.

And Mr. Trump misleadingly claimed that under Mr. Biden, “I guess 600,000, 670,000 lose their jobs.” That is an estimate of the effect of a nationwide ban on fracking from the U.S. Chamber of Commerce, but Mr. Biden has not said he supports such a ban.