How does one proceed to write an essay on the experience of time in Shakespeare?

You first want to gather quotes that show how Shakespeare looked at time. While you need to come to your own conclusions, one aspect of Shakespeare's thinking about time that might jump out at you is the way he contrasts chronological time to the way writing, in a sense, can stop time.
If you take a look at his sonnets in particular, you might notice that he feels he can immortalize his beloveds and keep them from chronological aging through his verses. He knows that in real time they will grow old, grow white hair, and die, but he argues that in his poems they stay young and alive forever.
Using this idea of "two times," you ought to be able to develop a thesis that might say something like, "Shakespeare argues in his poems that there are two kinds of time: chronological time and the time that is frozen through verse." Again, this is just a start: you will want to make a thesis that is your own. But once you have your thesis, you will want to support it through enough quotes to make your argument convincing.