What does "Alas! Poor Yorick. I knew him well," mean?

Yorick, in the scene you ask about (Act 5.1) in
Shakespeare's Hamlet, was more than someone who was once nice to
Hamlet, he was the court jester.  He bore Hamlet on "his back a thousand times" and was
kissed by Hamlet "I know not how oft." 

A court jester was
a clown who provided entertainment for the king and the royal household, but he was also
often a friend and confidant.  The role was one that allowed him to say whatever he
wanted to or thought necessary to the king, without fear of reprisal.  Perhaps, the
jester was the only person in the kingdom who could do
so. 

In Hamlet, Yorick apparently spent much time with
Hamlet when Hamlet was young. 

Importantly, notice that the
past has meaning for Hamlet here.  He has changed since his opening soliloquy in which
he compared the world to an unweeded garden (Act 1.2.135) and showed strong evidence
that he was suffering from melancholy, or depression.  All is not useless, now.  The
memory of Yorick has meaning for Hamlet.  And Hamlet misses
Yorick:

Where
be your gibes now, your gambols, your songs, your flashes of merriment that were wont to
set the table on a roar?  Not one now to mock your own grinning?  (Act
5.1.165-170)

Of course, in
addition to Hamlet revealing that he misses Yorick, here, his line of thought also
contributes to his past contemplations of existence, and forward to the contemplations
to come:

Dost
thou think Alexander looked o' this fashion i' th' earth?...And smelt so?  Pah!  (Act
5.1.175-78)

This is what
human life comes to, for a jester or a great conqueror.  But although Hamlet has not
stopped thinking and contemplating, he is not made inactive by his thinking.  In a
minute or two, he will leap into action and leap into the grave with Ophelia, and
declare:

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...This is I,

Hamlet
the Dane.  (Act
5.1.233-34) 

Hamlet has
changed in more ways than one.  From now on, as he says in Act
4.5.66:

My
thoughts be bloody, or be nothing
worth!