The political situation is extremely important. Renaissance culture tended to be organized hierarchically. Those higher in the so-called "great chain of being" had a special obligation to behave virtuously; any immorality on their part could bring not only personal damnation but also political unrest (or worse) for the entire nation. The fact that Claudius is such a corrupt and immoral king makes this play not only a personal tragedy for most of the major characters (especially Hamlet) but also a political tragedy for the Danish nation as well. The same selfishness and pride that have corrupted Claudius threaten the welfare of his people.
One part of the role played by the political situation is to add to the danger and suspense. During the time of the play, Denmark is not without enemy's. Fortinbras, the Prince of Norway, smolders against Denmark because Hamlet's father, the King of Denmark, killed his father, the King of Norway. This is definitely a political situation that adds intensity and suspense to the play. Also, Laertes is so outraged against the murder of his father, Polonius, that he throws aside his loyalty to the Prince of Denmark, Hamlet, in an attempt to have revenge. This is also definitely a political situation that increases the danger to Hamlet in Hamlet.
I was going to quote exactly the same line as #5. Note how the murder of the Old Hamlet and the evil that this represents spills over into all aspects of society in Denmark. Not only does it herald personal instability but also political instability as Denmark faces the possibility of an invasion. This of course creates a tense mood of oppressive uncertainty as men are working hard to prepare for war. It undercuts the confident impression of power and authority that Claudius presents us with in Act I scene 2 and likewise points towards deeper underlying issues.
"There is something rotten in Denmark" is one of the many famous lines from Hamlet. The political situation--that something "rotten"--is the catalyst for all Hamlet's soliloquies which, in turn, are what move the play forward. His disgust for Polonius, for Rosencrantz and Guildenstern, his mother, and, of course, for Claudius bring Hamlet to declare himself Prince of Denmark and battle for the kingdom. His death rids Denmark of it rottenness as the prince of integrity, Fortinbras, who has inspired Hamlet to action, becomes its ruler.
It sets the tone for the entire play. Not only does it give us history and how Fortinbras and King Hamlet's disagreement began, but it also foreshadows the violence to come in Denmark (Polonius' death, the deaths of Rosencrantz and Guildenstern, Ophelia's demise, and the final sword fighting scene where Gertrude, Claudius, Laertes, and Hamlet himself all die) as well as the violence and unrest that has already occured (King Hamlet's murder by the hand of his own brother). It also symbolizes the unrest of the internal conflicts Hamlet suffers from--his mother's too-quick marriage to his uncle, his birthright stolen out from under him, his father's death, and the appearance of his father's ghost.
It also serves with a convenient ending, as upon Hamlet's death, he announces Fortinbras the rightful heir to the throne of Denmark who, in turn, buries Hamlet with honors.
The political situation adds to the stress that is being felt by Hamlet and by Claudius. Everyone is kind of on edge because there is so much uncertainty. There are issues over who should be king. There are problems in Denmark's relationship with Norway. All of this plays a role in the play because it adds to the generally dark and unstable mood. The mood helps to create tension in our minds, which makes us more receptive to the tension and uncertainty that is going on in Hamlet's mind.
The country is in political unrest. The beloved King Hamlet is dead. Claudius is now at the head of the political system. He is a murderer. He is unjust. The country is in the hands of a man who is untrustworthy.
Claudius has killed his own brother. How can he be trusted? The political system is out of balance. This is clear in Act 1, Scene 2. Even Claudius admits that the country is disorganized in his brother's death, or so thinks young Fortinbras:
I will tell you now, as you know, young Fortinbras,
Not thinking very much of us,
Or thinking that our late dear brother's death
Made our country disorganized and no longer powerful,
There has to be some truth in this; otherwise, Claudius would not have brought up the idea. There is unrest and disorganization in the death of King Hamlet. Even the guardsmen admit that there is unrest:
That has purpose in it, which is no other,
As it seems to our country,
Than to recover from us, by war
And non-negotiable terms, those same lands
That his father lost, and this, as I understand it,
Is the main motive of our preparations,
The source of this our watch, and the chief reason
For this speed and commotion in the land.
The guardsmen are at watch in that another war does not break out. They admit there is commotion in the land.