How did the historical background of his age influence Shakespeare's writing ?
There is also historical reference that can be made in connection to particular plays.
The Tempest, for example, was probably based upon an actual shipwreck, and the play was definitely influenced by the discovery of the New World. The English were just beginning their experiments at colonizing America. Caliban, especially, appears as a character that is said to have been modeled on reports of the natives "discovered" in the New World. Shakespeare, it is believed relied upon published accounts of both the actual shipwreck and natives as source material for this play.
The writing of Macbeth was influenced by the taking of the English throne by James I, after Queen Elizabeth's death. He was the son of Mary, Queen of Scots (who some consider to have been Queen Elizabeth's arch rival) and had been on the Scottish throne before Elizabeth's death. This story of the rise to power of a Scottish king was written in honor of James.
Shakespeare's company gained from King James their highest honor, as well. King James became the company's patron and they were thereafter known as The King's Men. During Queen Elizabeth's reign they had been The Lord Chamberlain's Men.