The bumper proposal would include huge new social programmes and investment in the fight against climate change.
But it needs approval from Congress, where Republican Senator Lindsey Graham condemned it as "insanely expensive".
Under the plan, debt would reach 117% of GDP by 2031, surpassing levels during World War Two.
That would be in spite of at least $3tn in proposed tax increases on corporations, capital gains and the top income tax bracket.
Former President Donald Trump, a Republican, also ran up the deficit each year he was in office, and his final annual spending proposal had a price tag of $4.8tn.
The Biden budget includes a $1.5tn request for operating expenditures for the Pentagon and other government departments. It also incorporates two plans he has previously publicised: his $2.3tn jobs plan and a $1.8tn families plan.
Mr Biden, a Democrat, said his budget "invests directly in the American people and will strengthen our nation's economy and improve our long-run fiscal health".