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.