Look you, sir,
Inquire me first what Danskers are in Paris,
And how, and who, what means, and where they keep,
What company, at what expense; and finding
By this encompassment and drift of question
That they do know my son, come you more nearer
Than your particular demands will touch it.
At the top of Act II Polonius instructs Reynaldo to spy on his son, Laertes, in Paris. Polonius goes into great detail, explaining not only the types of information he wants, but also how Reynaldo should go about procuring that information. Although it isn't entirely clear what reasons Polonius has for spying on his son, his plot does demonstrate that he is a scheming and potentially treacherous figure.
I'll have these players
Play something like the murder of my father
Before mine uncle. I'll observe his looks.
I'll tent him to the quick. If 'a do blench,
I know my course.
In a long soliloquy in Act II, Hamlet announces his intention to use the troupe of players to set a trap for Claudius. He will instruct the troupe to perform a scene that reenacts Claudius' murder of the king. During the performance Hamlet will spy (or, as it actually happens in Act III, he will have Horatio spy) on Claudius to see if his reaction to the scene incriminates him. If Claudius "do blench" (i.e., flinch), Hamlet will know what to do next.
For we have closely sent for Hamlet hither.
That he, as 'twere by accident, may here
Affront Ophelia. Her father and myself,
We'll so bestow ourselves that, seeing unseen,
We may of their encounter frankly judge,
And gather by him, as he is behaved,
If 't be th' affliction of his love or no
That thus he suffers for.
Here Claudius explains to Gertrude his plan to spy on Hamlet and Ophelia. He and Polonius will hide and look on to see what Hamlet does when he arrives. They believe that if Hamlet doesn't know he's being watched, he'll act without pretense and give them a better sense of what's going on with him. More specifically, they believe his behavior will demonstrate once and for all whether or not his love for Ophelia has caused his "affliction" of melancholy and madness.
Let's further think of this,
Weigh what convenience both of time and means
May fit us to our shape. If this should fail,
And that our drift look through our bad performance,
'Twere better no assayed. Therefore this project
Should have a back or second that might hold
If this did blast in proof.
Claudius utters these lines to Laertes as they fashion their plot to kill Hamlet. Claudius emphasizes that he and Laertes will have to perform like actors, and he worries that if a "bad performance" should reveal their scheme, they will need some backup plan to ensure Hamlet's death no matter what. Following this quote, Claudius goes on to say that if Laertes does not manage to kill Hamlet in their duel, then a poisoned chalice of wine shall be offered to Hamlet following the match.