Who does the line, "Something is rotten in the State of Denmark," (Act I, Scene iv) refer to and who says it?

Since the Ghost is first seen by the watchmen, Shakespeare sets up the entire theme for this play in the quote spoken by Marcellus. He refers to King Claudius when he speaks of rottenness in Denmark, which can mean any kinds of moral evils and political corruptions.

Seeing the Ghost becomes a more developed point further in the play. The Ghost is, obviously, not a figment of Hamlet's imagination but an entity seen by others outside the royal house, for example, by the watchmen.

There is much "rotten in the State of Denmark" including the murder of King Hamlet by his brother, Claudius. Hamlet's mother, Gertrude, has wedded and bedded Claudius when the King is "but two months dead!"

Hamlet, then, is forced to deal with his father's murder and what he believes is his mother's adultery. At the close of the play, the 'rottenness' in Denmark leads to the demise of all the Royals and the end of the royal lineage.