Queen Elizabeth was a symbol and a very powerful figure in Shakespeare's time, and allusions to her permeates many cultural constructions of the time. You can read a thorough discussion of A Midsummer Night's Dream in relation to Queen Elizabeth in Louis Montrose's article (see the bibliography for details). Among other things, Montrose discusses Elizabeth in relation to contemporary ideas about amazons.
The 1998 film Elizabeth is amusing and gives some ideas as to what images of Elizabeth might have been like. There's more information about this film in the bibliography.