Hello, here is something I wrote for my Shakespeare studies. How is the grammar on this? Could anyone read over it and give me some feedback? Thanks!

What would you do?

I believe that the ghost that has appeared before me truly is my father’s spirit, and that everything it has said is true. The ghost did not flee before Horatio’s “cross,” and it shows Christian forbearance towards my “common” mother. Thus, I have no choice but to believe that this spirit truly is my father’s and not a devil trying to imitate him. I have little choice, then: I must avenge his spirit, and kill my uncle, Claudius! My father’s spirit calls to me, and I must avenge his foul murder!

This is not a selfish act of revenge, but an act of justice. Claudius is a tyrant: he killed a king, and worse, he killed his own brother, all to satisfy his own private ambitions. He is guilty of a sacrilegious attack upon the hierarchy and order instituted by natural law; by killing my father, Claudius destroyed this natural order, and it is my responsibility to right this wrong!

The killing of my uncle must be done quickly, and discretely. Claudius is no fool, and I am certain that the blood upon his hands will cause him to become paranoid; he will undoubtedly reach a stage where he deems it necessary to eliminate all threats in order to secure his power and I am certain that I would be his first target. So I must act first! I cannot poison my uncle or assassinate him in public. Such acts are too cowardly and too dangerous; a public murder could result in me being tried for treason.

The most discrete and most practical method of killing my uncle would be to stab him as he is sleeping with my “incestuous” mother. Being the prince means I can walk around the castle freely and carry my sword without arousing suspicion. At night, I could simply slip in, cover Claudius’ mouth to suppress his screams, and stab at his heart! Then, I can wake my mother and explain to her the reason for the assassination. This would fully comply with my father’s wishes, as not only would my uncle be dead, but my mother would not be harmed directly. All would soon be well.

The next day, I will deal with the court and the public. The latter would be easy to deal with, as I am loved by the people, and they will quickly accept me over Claudius. Further, the common people would not dare interfere in the affairs of kings; they will not make too much commotion, I am sure. The court, however, will be harder to convince and silence. Questions will be asked, and I will have to answer. Nevermind! I am the heir to the throne and the court could not, in reality, do anything. I could silence any who oppose my power; I am, after all, the rightful Prince of Denmark! I will soon be elected king and rule over my people justly; just as my father did, and just as he expects me to do.

I believe that the ghost that has appeared before me truly is my father’s spirit, and that everything it has said is true. The ghost did not flee before Horatio’s “cross,” [no quotation marks needed here] and it shows Christian forbearance towards my “common” [not quite the right word here, and why have you used quotation marks?] mother. Thus, I have no choice but to believe that this spirit truly is my father’s and not a devil trying to imitate him. I have little choice, then: I must avenge his spirit, and kill my uncle, Claudius! My father’s spirit calls to me, and I must avenge his foul murder!

This is not a selfish act of revenge, but an act of justice. Claudius is a tyrant: he killed a king, and worse, he killed his own brother, all to satisfy his own [repetition of own - redundant here, all your private ambitions are your own] private ambitions. He is guilty of a sacrilegious attack upon the hierarchy and order instituted by natural law; by killing my father, Claudius destroyed this natural order, and it is my responsibility to right this wrong!

The killing of my uncle must be done quickly, and discretely. Claudius is no fool, and I am certain that the blood upon his hands will cause him to become paranoid; he will undoubtedly reach a stage where he deems it necessary to eliminate all threats in order to secure [consolidate?] his power and I am certain that I would be [will be or am going to be] his first target. So I must act [strike?] first! [no exclamation mark here] I cannot poison my uncle or assassinate him in public. Such acts are [respectively?] too cowardly and too dangerous; a public murder could result in me being tried for treason. [the two parts of this sentence don't follow each other. I would use a colon after public, then full stop after dangerous and make the second clause a separate sentence]

The most discrete and most practical method of killing my uncle would be to stab him as he is sleeping with my “incestuous” [why quotation marks?] mother. Being the prince means I can walk around the castle freely and carry my sword without arousing suspicion. At night, I could simply slip in, cover Claudius’ mouth to suppress his screams, and stab at his heart! [stab him in/through the heart - no exclamation mark] Then, I can wake my mother and explain to her the reason for the assassination. This would fully comply with my father’s wishes, as not only would my uncle be dead, but my mother would not be harmed directly. All would soon be well.

The next [following] day, I will deal with the court and the public. The latter would be easy to deal with [repetition of deal with - appease/placate?], as I am loved by the people, and they will quickly accept me over Claudius. Further, the common people would not dare interfere in the affairs of kings; they will not make too much commotion, I am sure. The court, however, will be harder to convince and silence. Questions will be asked, and I will have to answer [them or which I will have to answer]. Nevermind! [Never mind is two words, it's not quite the right choice of phrase here, and no exclamation point] I am the heir to the throne and the court could not, in reality, do anything. I could silence any who oppose my power ; I am, after all, the rightful Prince [at this point, surely Hamlet is the rightful King rather than just the Prince?]of Denmark! I will soon be elected [by definition, kings are not elected] king and [will] rule over my people justly; just as my father did, and just as he expects me to do.

Looks pretty good to me. You have a lot of long and rather complex sentence constructions here (heavy on the semicolons!), but the declamatory tone seems quite appropriate.
The killing of my uncle must be done quickly, and discretely
The most discrete and most practical method of killing my uncle

I would advise you to check which word you want to use: discrete OR discreet. They are quite different in meaning.

I would also comment that stabbing someone with a sword might not be very easy to do if your other hand is over the victim's mouth.