How was Shakespeare connected to Queen Elizabeth the First?

Answer 1

She was Queen during part of his career as playwright/actor. She attended and approved many of his plays.

Answer 2

She was the Queen of England when he was writing. Her courtiers watched the plays and some of them became W.S's patrons (that means they paid for some of his expenses)

I do not believe that there were any other connections, but a good website would tell you.