How do you learn in the first scene that something is wrong in Denmark in Act 1 of Hamlet?

There are actually two distinct problems in Denmark that
we learn a little about in scene 1.  Typical of Shakespeare, he establishes the mood and
the initial conflicts in the first scene, and then has the details of the story unfold
in the subsequent scenes in Act 1.

The first problem that
we see is that there has been a ghost wandering the castle grounds for a couple of
nights.  The guards know what they have seen, but have asked Horatio, a school friend of
Hamlet's to come out and verify it.  Horatio recognizes the appearance of the ghost to
resemble the recently deceased King Hamlet.  He tries to communicate with it, but the
ghost merely stalks away.  Horatio tells us the appearance of a ghost can have no good
cause.  Ghosts come back to reveal a secret; to finish unfinished business; or to reveal
treasures. Even worse, the ghost could be a devil in the disguise of a loved one and out
to damn the soul of a living person by enticing them to commit some foul act.  The
arrival of a ghost is never really a good thing according the Renaissance understanding
of ghosts.

The second thing that is wrong in Denmark is
that Prince Fortinbras of Norway has "sharked up a list of lawless resolutes" --
mercenary soldiers -- to attack Denmark in the hopes of his regaining lands that his
father lost in a battle with King Hamlet many years before.  This external threat to
Denmark is the reason, Horatio reports, for the increase in military preparations going
on seven days a week. 

It is through the minor character of
Horatio that we learn about these over-arching conflicts for Denmark.  By the end of Act
1 we know a lot more what the ghost represents, and the play then progresses through
rising action to the climax and resolution of each of these
conflicts.