How does the murder of King Hamlet effect the country? (corrupt it, essentially)I'm doing an essay on how the gov't (C,H,G&P) did corrupted things,...

With the death, or murder to be precise of King Hamlet, the country is made vulnerable to attack from Norway, particularly from young Fortinbras who is anxious to regain land that his father lost to King Hamlet. 

The death of a rightful King, usually opens up opportunity for challenge by enemies who have remained silent, to emerge and cause conflict, it is a catalyst for the unleashing of powers within the universe that force or restore the natural balance, that is the case in Hamlet. 

It is especially true in Shakespeare when a rightful king is unjustifiably removed from the throne through an act of murder, like in the case of King Duncan being murdered by Macbeth where Scotland suffers because of Macbeth's crimes.  Similarly, when King Hamlet is murdered, the rules that govern reality are suspended, representing the immorality of Claudius's deeds.   

King Claudius fears an attack and contacts the King of Norway, the Uncle of Fortinbras, and advises him to keep his nephew calm and under control. There is external turmoil that threatens Denmark.  The country might appear vulnerable to attack because they have had a transition of power and are technically grieving their dead monarch.  Claudius is quick to establish that Denmark is strong and ready to defend itself from attack.

"He informs the court that Fortinbras has been pestering him to surrender the land and may think that Denmark may not be ready for war because they have been in mourning or because they are weak. The King sends Cornelius and Voltemand to carry a message about Fortinbras' actions to Old Norway, Fortinbras' uncle, who is ill and bedridden. The King of Norway is in the same position as this King: they are both brothers to the last Kings and have nephews."

Then there is internal turmoil that threatens the country in the mysterious nature of the death of King Hamlet.  Hamlet is visited by his father's ghost and instructed to avenge his death, which was murder.  There is a strong sense of tension created by Hamlet's pretense of madness, which is a good cover for his real feelings of rage, injustice and unresolved grief over the loss of his father.  His quirky, odd behavior sends shock waves through the court. The more Hamlet acts like a lunatic, the calmer King Claudius looks to his wife, Hamlet's mother Queen Gertrude.  

However, in the court of King Claudius, there is a great deal of chaos caused by Hamlet's odd behavior which seems to take on a life of its own with its own destructive forces, separate from Hamlet's plans to avenge his father's death and make his mother look into the face of her betrayal of her rightful husband.