In the beginning of Shakespeare's Hamlet, we learn several things about the Danish prince.
We discover that Hamlet is a traditional young man of character. When he returns home for his father's funeral, one of the first things the reader finds is that Hamlet is disgusted at the speed with which his mother has remarried. Marrying her brother-in-law, by Elizabethan standards, is considered incest.
Even though Claudius and Gertrude are related only through marriage, the union between a woman and her husband's brother...was considered incest…[and] was explicitly forbidden by the Catholic and Anglican faiths.
When his schoolmate, Horatio, tells Hamlet that he has come to the castle for Old Hamlet's funeral, Hamlet sarcastically notes that Horatio must be enjoying at joke at Hamlet's expense: he says Horatio must have really come for his mother's wedding. He exaggerates, pointing out that the leftovers from the funeral dinner were saved and used for his mother's remarriage to Claudius. In this way, he was observing that the one followed the other much too quickly—that the "o'er hasty" marriage was in poor taste, in his opinion.
My lord, I came to see your father's funeral.
I prithee do not mock me, fellow student.
I think it was to see my mother's wedding.
Thrift, thrift, Horatio. The funeral baked meats
Did coldly furnish forth the marriage tables. (I.ii.181-186)
Hamlet is so devastated by his father's death that he wishes he could die: this lets the reader know how dear his father was to him. We also know that he is a religious man because he knows that to kill himself would be a mortal sin, and he alludes that he will not do so:
O, that this too too sullied flesh would melt,
Thaw and resolve itself into a dew,
Or that the Everlasting had not fix'd
His canon 'gainst self-slaughter! O God! God! (132-135)
When his mother asks him why Hamlet seems so depressed, he is emphatic that he does not "seem" so crushed. He is not pretending, but is deeply saddened by losing his father. Hamlet's beliefs also reflect those of the supernatural that Elizabethans had during Shakespeare's life.
The supernatural is used abundantly because of popular belief and demand from the people.
When Horatio notes that he has seen a spirit that looks like Old Hamlet, Hamlet immediately arranges to go to the battlements to see if the ghost really is that of his father.
While Hamlet says that the spirit is "honest" when the Ghost exposes his murder at Claudius' hands, he will still search for proof. Once again, Hamlet shows the influence of the Church during Shakespeare's time: he must be certain that Claudius has murdered his father before he takes action, or he will forfeit his soul to the devil. So Hamlet, cleverly, decides to pretend to be crazy until he gathers the information he needs to prove that Claudius is a murderer.
At the start of the play, the audience learns that Hamlet is a man of integrity appalled by his mother's "incestuous" marriage; he is also a man who follows the tenets of his faith, is a devoted son, and is clever enough to cover his tracks as he investigates the credibility of what the Ghost has told him. His actions are defensive; he is not unkind or malicious, but dedicated to avenge his father's death.
Hamlet is a character that does not actually appear in Act I scene 1. Like other characters in other plays, such as Macbeth, he is only introduced by others, heightening our anticipation of actually meeting the central protagonist and presenting the audience with various impressions and ideas about his character before he finally emerges on the stage. Act I scene 1 actually talks more about the old King Hamlet than his son. Horatio talks of King Hamlet's bravery and prowess in battle and of his recent death. He only mentions his son towards the end of the scene when he thinks about what to do about the Ghost:
Let us impart what we have seen tonight
Unto young Hamlet; for upon my life,
This spirit, dumb to us, will speak to him.
Horatio therefore presents the audience with an idea of Hamlet that suggests that he will be able to communicate with this supernatural visitation in a way that he has not been able to. Also, the audience perhaps will predict that Hamlet, being King Hamlet's son, will share some of the same traits as his brave and noble father. What this scene does is to raise the anticipation of the audience and to increase their curiosity as they wait to meet Hamlet himself and to see what he is like.