The scene opens with Claudius and Gertrude questioning Rosencrantz and Guildenstern about Hamlet’s state of mind. As this part winds down, Claudius has an aside that reveals his feelings. An “aside” is something that only the audience can hear:
How smart a lash that speech doth give my conscience! The harlot’s cheek, beautied with plast’ring art, Is not more ugly to the thing that helps it Than is my deed to my most painted word. O heavy burden!
The king is expressing feelings of guilt over his actions (killing his brother, Hamlet’s father). He’s saying that no matter how he tries to smooth things over (his “painted word”), he is still defined by what he has done (“my deed”). This is a type of conflict called “internal conflict.” That means a character experiences the conflict inside himself—it is not the direct result of someone or something else (although it is often the indirect result of someone or something else).
Next, we have a monologue from Hamlet. This speech boils down to the eternally famous first line:
To be, or not to be: that is the question.
Hamlet is now expressing his internal conflict. Should he keep on living in his misery (to be), or just give up and end it all (not to be). Where the king was internally conflicted with guilt, Hamlet’s conflict is with indecision. What to do, what to do?
Finally, we have an external conflict, which is a problem a character experiences outside themselves, with another person or entity of some type. In this case, Hamlet and Ophelia conflict.
I did love you once.
Indeed, my lord, you made me believe so.
You should not have believed me, for virtue cannot so inoculate our old stock but we shall relish of it. I loved you not.
I was the more deceived.
Hamlet is cruel to Ophelia in this scene, telling her that he never truly loved her and that she should go live in a nunnery.
In Act three, Scene one, King Claudius thinks Hamlet is showing signs of craziness. He, Gertrude, Polonius and Ophelia are having a conversation about whether or not Hamlet is losing his sanity. They set up a meeting between Ophelia and Hamlet to learn whether or not Hamlet is love sick. While Hamlet and Ophelia have a conversation, Claudius and Polonius are hiding in the background to learn what is the matter with Hamlet.
Hamlet is having his own personal conflict. He is questioning whether his suffering is to be or not to be. In his famous soliloquy, Hamlet questions whether he is to retaliate against his father's murderer or whether it is more noble to suffer in the mind:
To be, or not to be, that is the question.
Is it nobler in the mind to suffer
The slings and arrows of outrageous fortune
Or to fight against a sea of troubles,
And end them by fighting?
Hamlet is clearly in conflict within himself. When he and Ophelia converse, he insists that he does not desire to be with her. He insists that she get herself off to a convent:
Get yourself to a convent. Why would you give birth to
sinners? I don’t care about my sincerity, but I could
accuse myself of such things that it were better my
mother had not had me. I am very proud, revengeful,
ambitious, with more offenses at my command than I
have thoughts to put them in, imagination to give them
shape, or time to commit them in. What should such
fellows as I do crawling between earth and heaven? We
are wicked men, all, believe none of us. Take yourself to a convent.
After this conversation, Claudius decides that Hamlet is not mad. He is just troubled. Of course, Claudius has no idea what Hamlet is feeling within himself about the murder of his father.
The scene ends with Claudius discussing Hamlet's condition with Polonius:
Love! his affections don’t go that way,
And what he said, though it was a little unorganized,
Wasn’t like craziness. There's something in his soul
Over which his depression sits like a bird on an egg,
And I doubt that egg will hatch and the “bird”
Will be something dangerous, which to prevent,
I have quickly decided
To write it down like this. He shall go quickly to England
To collect money due to us.
Happily the seas, and different countries
With other things, shall cure and get rid of
This problem in his heart that has such a hold on him,
Which his still active brain makes him like this
From being himself. What do you think about it?
Indeed, Claudius plans to get rid of Hamlet. He does not know what to do with a Hamlet who is so depressed.