There’s a divinity that shapes our ends,
Rough-hew them how we will—
 

Hamlet utters this quote early in the final scene of the play as he tells Horatio about how he got revenge on the disloyal Rosencrantz and Guildenstern by replacing a note that Claudius had written to the King of England requesting that he, Hamlet be killed upon his arrival with a forged note requesting the the executions of Rosencrantz and Guildenstern instead. In the quote, Hamlet talks about how he was able to pull this off by using instinct rather than intellect, essentially saying that God will guide us no matter how often we screw up.

The second half of the explanation for this quote in Quotes by Character: Hamlet (the tenth and final quote) discusses a crucial change in Hamlet’s character after he accidentally kills Polonius—how his actions go from being deliberative and indecisive to impulsive and direct—and how this change in approach will prove decisive in his upcoming final encounter with Laertes and Claudius.

Was 't Hamlet wronged Laertes? Never Hamlet.
If Hamlet from himself be ta'en away,
And when he's not himself does wrong Laertes,
Then Hamlet does it not. Hamlet denies it.
Who does it, then? His madness.

With this quote in which he refers to himself in the third person just prior to his deadly fencing match with Laertes, Hamlet denies responsibility for killing Polonius on the grounds of his madness. This quote is further discussed in Quotes by Theme: Madness versus Cunning (the sixth quote).

He’s fat, and scant of breath.—
Here, Hamlet, take my napkin, rub thy brows.
The queen carouses to thy fortune, Hamlet.

 Scholars have long debated the meaning of this quote from Gertrude about her son Hamlet as he fences with Laertes in the death-filled final scene of the play—particularly about what she means when she describes her son as “fat.” While the word did mean “overweight” in Shakespeare’s time, it also had a secondary meaning of “sweaty,” which would seem to make more sense, since she offers Hamlet a napkin after she calls him “fat.” The final line of the quote (“The queen carouses thy fortune, Hamlet”) is more direct. It basically means that she is drinking to his good luck and happiness. But her drink is laced with poison, so she dies soon after toasting Hamlet.

Horatio: I am more an antique Roman than a Dane.
Here’s yet some liquor left.
(lifts the poisoned cup)
Hamlet: As thou'rt a man,
Give me the cup. Let go! By heaven, I’ll have ’t.
(takes cup from HORATIO)

In this exchange between Horatio and Hamlet just before Hamlet dies in Act 5, Scene 2, Horatio contemplates suicide by drinking the poisoned liquor that has already killed Queen Gertrude, but Hamlet stops him. Horatio’s comment, “I am more an antique Roman than a Dane” is a reference to Romans of antiquity frequently dying by drinking poison (as seen in Shakespeare’s own play Julius Caesar) in contrast to the presumedly Danish custom of dying in battle. Shakespeare included many links to Rome and to Julius Caesar, (which was staged few years earlier and would be well known to the audience) in Hamlet, starting in its first scene, when Horatio compares the appearance of the Ghost to omens that preceded Caesar’s assassination.

Moreover, Shakespeare frequently used Latin and Italian names in his plays regardless of their settings. Examples of this in Hamlet include Claudius (the Danish king shares his name with the Roman emperor who conquered Britain), Polonius (who proudly mentions that he once played the role of Julius Caesar in Act 3, Scene 2), Laertes, the watchmen Marcellus and Bernardo, and, of course, Horatio, who proves to be Hamlet’s only true friend in the play. Since Latin Horatio’s name roughly translates to “keeper of time” in English, it is fitting that Hamlet stops Horatio from committing suicide so he can at survive the play and carry his friend’s story forward.

O, I die, Horatio.
The potent poison quite o'ercrows my spirit.
I cannot live to hear the news from England.
But I do prophesy the election lights
On Fortinbras. He has my dying voice.
So tell him, with th' occurrents, more and less,
Which have solicited. The rest is silence.
O, O, O, O. (dies)

Right before he dies, Hamlet asks Horatio to tell everyone the truth about what has happened. Then he delivers these, his final words, which focus on Denmark’s future as specifically asks Horatio to tell Fortinbras that he approved of Fortinbras taking control of Denmark and its crown. You can read more about Hamlet’s famous last words, “the rest is silence” in Quotes by Theme: The Impact of Uncertainty (the fourth quote).

Let us haste to hear it,
And call the noblest to the audience.

These words are spoken by Fortinbras near the end of the play after Horatio has explained to him what has happened, as Hamlet requested before dying, and Fortinbras has ordered that Hamlet’s body be carried “to the stage.” The explanation to this quote in Quotes by Theme: Performance versus Reality (the fifth quote) discusses the significance of Fortinbras’s using theatrical language and how up to the very end, Hamlet is a play about the blurring of the lines between pretending and the actuality of real life.