What is the main idea of Hamlet's "O all you host of heaven" soliloquy in Act 1, Scene 5?

In answering this question, let's first of all look at the context in which Hamlet makes his speech. The ghost of Hamlet's father has just appeared to him and told him of the manner of his death.

Hamlet's father, old King Hamlet, was asleep in the garden one day when his brother Claudius crept up and poured poison down his ear. Hamlet has never liked or trusted Claudius and perhaps suspected that his uncle/stepfather had something to do with King Hamlet's murder. But now he has the proof from beyond the grave, and, as one can imagine, he's absolutely furious at what he hears.

In his speech in Act 1, Scene 5, Hamlet vows revenge on Claudius for his murderous misdeeds. He ends the speech with the words “Now to my word,” which means that it's time to make good on the vow he made to his father.

The dramatic significance of this vow is not hard to detect. Hamlet's determination to gain revenge on Claudius will dominate the rest of the play and provide motivation for most of his actions. From here on in, the play is primarily concerned with Hamlet's planned revenge, even though he takes an awfully long time putting it into practice.

In this soliloquy, spoken immediately after the spirit of Hamlet's father has vanished, Hamlet takes a moment to reflect on what he has just learned. Part of this soliloquy is simply Hamlet's rage and shock at his uncle's treachery and the fact that his mother married such a man. He also resolves to dedicate himself completely to avenging his father's murder:

All saws of books, all forms, all pressures past,
That youth and observation copied there; 
And thy commandment all alone shall live 
Within the book and volume of my brain, 
Unmix'd with baser matter.

One of the more fascinating complexities to this play is that Hamlet is unable to do what he swears to do in this soliloquy. He is distracted, mostly, by his own introspective nature, which causes him to reflect on many of the events that occur around him rather than acting on them immediately. He also commits himself to determining the veracity of the ghost's accusation, which he does with the play. 

Finally, Hamlet resolves himself to remember one crucial thing which will serve him well, particularly in his dealings with Guildenstern and Rosencrantz: 

That one may smile, and smile, and be a villain; 
At least I am sure it may be so in Denmark.

What is the dramatic significance of Hamlet's "O all you host of heaven..." speech in act 1, scene 5?

One of Hamlet's briefer soliloquies, beginning "O all you host of heaven," occurs in act 1, scene 5, when the ghost of his father has just left after having described how he was murdered and exhorted Hamlet to avenge his death. This speech is an emotional reaction to his father's revelations and a passionate commitment to focus completely on the task of revenge.

This speech is dramatically significant because it shows Hamlet at the peak of his determination and certainty. He says that he will "wipe away all trivial fond records" from his mind so that the only thing remaining in it will be his father's commandment.

Before this scene, Hamlet has been hostile and suspicious in his attitude toward Claudius and Gertrude; however, until now, he has not been certain of his uncle's guilt and therefore has not resolved to take any specific action against him.

After this point in the play, he appears to waver in his revenge mission several times. The audience is never quite sure whether he is intentionally biding his time or procrastinating, whether he is pretending to be mad or actually so. Immediately after hearing his father's story, however, Hamlet gives an immediate and uncomplicated expression of his resolve to be avenged.

What is the significance of the quote "O all you host of heaven" in Hamlet?

In act 1, scene 4 of Shakespeare's Hamlet, the Ghost of Hamlet's father appears to Hamlet, Horatio, Marcellus, and Bernardo, but he doesn't speak to them. Instead, in scene 5, the Ghost leads Hamlet away from the others to another part of the castle.

The Ghost tells Hamlet that he only has a short amount of time before he has to return to Hell, and he tells Hamlet to pay close attention to what he has to say.

The Ghost then tells Hamlet that it was Claudius, Hamlet's uncle, who killed him, and demands that Hamlet swear to avenge his "foul and most unnatural murder" (1.5.29).

The Ghost tells Hamlet how Claudius killed him, and leaves Hamlet with the words, "Adieu, adieu, adieu! Remember me" (1.5.96).

Hamlet falls to his knees, and in a short soliloquy he calls on Heaven, earth, and even Hell to help him keep his promise to his father's ghost and to help him seek revenge against Claudius:

O all you host of heaven! O earth! What else?
And shall I couple hell? O, fie! Hold, hold, my heart!
And you, my sinews, grow not instant old,
But bear me stiffly up. Remember thee? (1.5.96–100)

Hamlet vows not only to remember his father and what his father told him but to devote his life to avenging his death:

I'll wipe away all trivial fond records,
All saws of books, all forms, all pressures past,
That youth and observation copied there;
And thy commandment all alone shall live
Within the book and volume of my brain...

Now to my word:
It is ‘Adieu, adieu! Remember me.’
I have sworn't. (1.5.104–117)

Horatio, Marcellus, and Bernardo have been searching for Hamlet, and when they find him, he demands that they swear on his sword not to tell anyone about what they've seen or heard. With prompting from the Ghost under their feet telling them to "Swear!", they swear.