First, it must remembered that Shakespeare was playwright, NOT a historian. He used the history plays to make points about war and power. The end of the Plantagenet family as the rulers of England was filled with both internal and external wars, the War of the Roses and The Hundred Years War.
The cycle begins with Richard II, who by most accounts was not a very good king. He came to the throne as a child and never learned how to rule. He misused his power and lost his throne as a result to his cousin, Henry IV, who was never comfortable on the throne since he usurped it. Henry V followed his father and became a hero with his victory at Agincourt. However, he died young and was succeeded by his infant son, another minority king. He was a good man but not a very good king. After losing in France, Henry VI was forced to fight for his throne at home, the War of the Roses. The trilogy ends with Richard III and his defeat at Bosworth Field and the rise of the Tudors.
In telling his story of this turbulent history, Shakespeare twisted the history to suit his dramatic purpose. For example, at the end of Henry VI, Shakespeare has the young Richard kill the king. It is a wonderfully dramatic scene but it didn't happen. When Henry VI died, Richard was a young boy living in France. He did not fight in the war. Shakespeare aged the character so he could be established him as a villain. In reality, Richard III was a good king and probably had nothing to do with the deaths of the little princes since they would not have been recognized by the church as the the heirs to the throne due to a previous marriage arrangement for Edward III. Edward had been pledged to marry another woman and was still bound to her when he married Elizabeth Woodville.
It must also be remembered that the Tudors came to the throne by defeating Richard III, the last Plantagenet king. It must also be noted that the Tudor came from an illegitimate line from John of Gaunt. Elizabeth was a Tudor. If Shakespeare knew the truth he was wise enough not to tell it. For him. history was a tool he could use to manipulate to tell his story.