In act 3, what is the purpose of the scene with Horatio?

In Act 3, scene 2, Horatio, as luck would have it, happens to stroll in just as the players are about to perform the "Mousetrap" for the court. Hamlet greets Horatio and talks about how he likes him for being so calm and level-headed, because it means he has good judgment. He then asks Horatio to watch Claudius very carefully to see how he reacts to the play, which will reenact the murder of Hamlet's father just as the ghost described it. Hamlet says:

Observe mine uncle. If his occulted guilt
Do not itself unkennel in one speech,
It is a damnèd ghost that we have seen,
And my imaginations are as foul
As Vulcan’s stithy.

What Hamlet means in that speech is that either Claudius will react with the guilt to the murder scene or he won't. If he doesn't, that indicates that the ghost is "damned," sent by Satan to try to tempt Hamlet. A lack of reaction also suggest that Hamlet's imagination is "foul" for thinking his uncle a murderer.
Hamlet says they should compare notes after the play is over. Horatio agrees. Once again, we witness Hamlet behaving in a very methodical fashion. He wants a trustworthy second person to confirm whether Claudius's actions shows guilt or innocence.

In act 3, scene 2, of Hamlet, the purpose of the dialogue between Hamlet and Horatio is to reinforce the guilt of Claudius in the murder of Hamlet's father, the King. Prior to their dialogue, Gertrude and Claudius have been watching the play Hamlet calls "The Mouse-trap" that shows the player Lucianus pouring poison into Gonzago's, the sleeper's, ear. This is a reenactment of the murder of Hamlet's father by Claudius.
Hamlet has planned the performance of this play to see if Claudius's reaction will be one of guilt, and it is. After the "imitation" death, Claudius is very disturbed and rushes away. Hamlet and Horatio are left alone, and Hamlet asks Horatio if he "didst perceive" Claudius's reaction. Horatio replies that he did "well note him." In this way, Horatio acts as a witness to the guilt of Claudius in the murder of the King.