To the best of my knowledge most people in the Elizabethan era (the late 1500's) did not actually believe in Greek mythology.
The vast majority of English people in the Elizabethan era were either Protestant Christians (like the Queen) or Roman Catholics (whom the Queen allowed to worship as they pleased). Neither Protestant nor Catholic theology ascribes to the belief in multiple, human-like gods such as those found in Greek mythology.
Nonetheless, the stories of Greek mythology were well known to most Elizabethans, particularly those who were well-educated. To be considered well-educated, one needed to have a thorough knowledge of the major characters in Greek mythology. People looked to these stories as important creations that contain great truths about human nature. That is why authors like Shakespeare make so many references to stories from Greek mythology.