Shenanigans!

Elizabeth I (7 September 1533 – 24 March 1603), reigned from 17 November 1558 until her death.
James VI & I (19 June 1566 – 27 March 1625) became King of England on 24 March 1603 and ruled until his death.
William Shakespeare William Shakespeare (baptised 26 April 1564; died 23 April 1616) has a gaping hole in the middle of what should be his career, but is accounted for by 1592 with plays in production in London.
Christopher Marlowe (baptised 26 February 1564; died 30 May 1593)first play performed on the regular stage in London, in 1587, was Tamburlaine the Great.

So, in order:

Elizabeth assumes the throne in 1558.

Kit Marlowe's first play performed on the regular stage in 1587.
William Shakespeare's re-emergence into history shows plays running in 1592. He is, however, still acting.
Kit Marlowe dies 1593.
Elizabeth dies in 1603. James assumes the throne. Hamlet, MacBeth, and King Lear all see production after 1603.
Shakespeare is assumed to have left acting around 1605.
No plays are attributed to Shakespeare after 1613.
Shakespeare dies 1616.

Shakespeare was not Elizabeth's playwright. Marlowe was. Shakespeare belonged to James.

And then there's Thomas Kyd: http://en.wikipedia.org/wiki/Thomas_Kyd

And if you want a good Marlowe read, try this one: http://en.wikipedia.org/wiki/Doctor_Faustus_%28play%29