Who said "Something is rotten in the state of Denmark" and "Heaven will direct it" in Shakespeare's play Hamlet?

This conversation occurs between Horatio and Marcellus. 

Horatio is Hamlet's friend and Marcellus is one of the watchmen.  Horatio and Marcellus are reacting to the appearance of the ghost and Hamlet’s reaction to him.  The ghost is kind of a mystery at first.  They can’t get him to stick around and can’t talk to him.  Hamlet is a little baffled, after all this is the ghost of his father.  The ghost beckons to him, and he follows it. 

HAMLET

My fate cries out,
And makes each petty artery in this body
As hardy as the Nemean lion's nerve.
Still am I call'd. Unhand me, gentlemen.
By heaven, I'll make a ghost of him that lets me!
I say, away! Go on; I'll follow thee.

Exeunt Ghost and HAMLET (Act 1, Scene 4)

Seeing ghosts roaming around your castle would be disturbing, especially the ghost of your dead king.  In a way, something is rotten in Denmark.  Claudius killed King Hamlet, and then married his wife.  Hamlet is incredibly upset about this, because he feels that his mother married too soon and his uncle should not have married her in the first place.

Horatio seems to have faith in Heaven that Demark will be okay, while Marcellus is not taking any chances.  Hamlet is talking to a ghost, after all.  The conversation between Marcellus and Horatio is important because it demonstrates the general state of unease in Denmark due to recent events.  They are not all that surprised that the ghosts are getting involved.  Denmark seems to be in trouble.

The ghost will direct the action of the play, actually.  He insists to Hamlet that he get revenge due to Claudius’s betrayal of him.  This puts Hamlet in an uncomfortable position.  His decision to feign madness so he can keep an eye on everyone just seems to reflect Marcellus’s comment about how Denmark is rotting.