How are Horatio, Hamlet, Claudius, the Ghost, Laertes, Gertrude, Ophelia, and Polonius connected in Act One of Shakespeare's Hamlet? In other words, how would you categorize the alliances in Denmark's royal court?
In Act One of Hamlet by William Shakespeare, there are several camps or alliances at work.
In Act One, several things are apparent. Hamlet's father has died and he is back at Elsinore for the funeral, which has quickly become the wedding between his uncle, Claudius, and his widowed mother, Gertrude. When Hamlet's school friend, Horatio arrives, it is clear how Hamlet feels about this turn of events:
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.
Indeed, my lord, it followed hard upon.
Thrift, thrift, Horatio. The funeral baked meats
Did coldly furnish forth the marriage tables. (I.ii.181-186)
Sarcastically, Hamlet notes that the wedding took place so soon after the funeral, that the leftover food from the funeral could well have been served to the wedding guests (though this is an exaggeration: it has been two to three months since the funeral, still a scandalously short time). However, Gertrude and Claudius urge Hamlet to put off his mourning and cheer up, but Hamlet is having none of it. So things in Hamlet's family are not going "well."
Also in Act One, Laertes has asked permission of the new King to leave the castle. As he is ready to depart, Polonius shares words of wisdom with his son (which, ironically, he is not wise enough to follow himself); Laertes also warns Ophelia about her romantic relationship with Hamlet. Laertes notes that nothing good can come of it, as Hamlet is royalty and she is not—so she should keep her distance, for Hamlet's selection of a wife is a matter of state, not of matter of love:
For he himself is subject to his birth.
He may not, as unvalued persons do,
Carve for himself; for on his choice depends
The safety and health of this whole state,
And therefore must his choice be circumscribed
Unto the voice and yielding of that body
Whereof he is the head. (I.iii.21-27)
At the same time, we can infer from this conversation that Hamlet and Ophelia have been seeing each other, though it is not until Act Two that the reader (or audience) will see them interact.
Another connection between the royal family and Polonius' family is that Polonius serves the King. He is...
...an elderly and long-winded courtier and chief counselor in the Danish court. Polonius demonstrates a propensity for hypocrisy and spying...
Horatio and Hamlet are school friends. However, it is also Horatio that brings news of the appearance of a specter looking very much like Old Hamlet (Hamlet's dead father). In scenes four and five, Hamlet goes to the battlements with Horatio and others to confront the Ghost in order to see what it wants.
The Ghost seems to be Hamlet's dead father. He appears to his son and commands—if Hamlet loved his father—that he avenge his murder by Claudius, something that shocks and disgusts Hamlet:
List, list, O, list!
If thou didst ever thy dear father love—
… Revenge his foul and most unnatural murder. (I.v.26-27, 29)
Murder most foul, as in the best it is;
But this most foul, strange, and unnatural... (30-32)
...The serpent that did sting thy father's life
Now wears his crown. (44-45)
O my prophetic soul! My uncle!
The characters' interactions overlap, moving beyond family relationships, that of friends and even of employees. They are all intertwined, the actions of many affecting the actions of most of the others.