What is the problem that Hamlet faces in Act I of the play?Provide quotes from the play.

Hamlet's primary problem in Act 1 is his hearing the news
that the ghost of his dead father has been seen at the castle, and then his subsequent
meeting with the ghost.  The mere presence of a ghost is a bad omen -- ghosts don't come
for any good reasons.  Marcellus is prompted to say, "something is
rotten in the state of Denmark."  Marcellus doesn't even know that the ghost is coming
to reveal, but it can't be good. 

Hamlet's immediate
problem with the ghost is that this ghost could just be a devil in the disguise of a
loved one who "might tempt you towards the flood or the summit of cliff."  Horatio is
cautioning Hamlet to act with care because the ghost could be dangerous.  Hamlet won't
be deterred, and he leaves the others to go hear what the ghost has to
say.

During this scene, many of Hamlet's worst imaginings
come true.  The ghost reveals that he died with sins on his soul and is suffering in
purgatory -- a place where he is "confined to fast in fires" and that is so awful that
to hear more of it would make young Hamlet "harrow up thy soul, [and] freeze thy young
blood."

Then the ghost gets to his main purpose.  He
reveals that Claudius murdered him in the garden while he was sleeping, hence "the
serpent that did sting thy father's life / Now wears his crown."  The ghost of King
Hamlet expresses his distress over this, as well as, his disappointment in Gertrude's
quick remarriage to that murderer, Claudius.  He says to Hamlet, "Let not the royal bed
of Denmark be / A couch for luxury and damned incest."  He is asking Hamlet to seek
revenge.  He specifically tells Hamlet to focus on Claudius, and to "taint not thy mind,
nor let thy soul contrive / Against thy mother
aught." 

After all of these revelations, it is clear that
Hamlet's biggest problem in Act 1, and going from there, is the request by his father to
avenge his death and make right the moral situation in Denmark.  It is quite a daunting
task, and it takes Hamlet the rest of this very long play to fulfill all of his duties
to this situation.