What is the purpose of the Yorick speech?

It shows Hamlet's growing preoccupation with death. In particular, it illustrates how it makes fools of us all, bringing each and every one of us down to the same level. Yorick was once a professional fool, "a fellow of infinite jest, of most excellent fancy." Yet now look at him. He's just a skull full of dirt, decaying in the grave.
Hamlet uses the example of the deceased court jester to reflect on the transience and impermanence of our lives upon this earth. It doesn't matter who we are or what we do in life, we will one day leave this earth behind for good. In the democracy of the dead, we are all equal; Yorick's skull is virtually indistinguishable from Alexander the Great's. We can try and distract ourselves from our inevitable end—this is what Yorick did with all this tricks, jokes and songs—but distractions they remain. They cannot substitute for a frank, courageous confrontation with our own inescapable mortality.