According to most critics, the source material for most of Shakespeare’s histories (including The First Part of Henry the Sixth) was Raphael Holinshed’s Chronicles of England, Scotland and Ireland. Holinshed was only one of a three main authors of the work (the other two being William Harrison and Richard Stanyhurst), and their work was first printed in 1577, about fifteen years before the composition of 1HenryVI.
While there were three authors to the Chronicles, when most critics discuss Shakespeare’s source, most point to Holinshed, who worked on the sections about pre-Norman England (which includes stories that spawned King Lear [800 B.C.], Cymbeline [first century A.D.], and Macbeth [the decades before the Norman Invasion]), as well as the English line of kings from William the Conqueror to Elizabeth I (which would include source material for all the other English histories).
Beyond Holinshed’s history, it appears that Shakespeare might have also used Edward Hall’s The Union of the Two Illustrious Families of Lancaster and York; some also believe that Samuel Daniel’s long poem “The First Four Books of the Civil Wars” might have also been an influence, but given its publication date of 1595, this seems doubtful.
Those who have read the Chronicles (a membership that does not include me, though I would love to [when I can lay my hands on a readable copy… and I get a boatload of free time…]) tend to believe that much of Act Two (particularly the Rose Briar scene) are purely fictitious.