What are some ideas about the relationship between Hamlet and Claudius and between Hamlet and Laertes in Hamlet?
I'm having trouble thinking of ideas for an essay.
You might consider writing an essay in which you compare Hamlet's relationships with Claudius and with Laertes. There are some definite similarities. For instance, Hamlet's relationships both of them were troubled from the beginning of the drama and then deteriorated further, ending in death for all three.
In the beginning of the play, Hamlet and Laertes are known to each other, but they are not friends. Laertes has no respect for Hamlet; he warns Ophelia about Hamlet, believing that Hamlet is dishonest and predatory in his feelings for Ophelia. His distrust and distaste for Hamlet become outright hatred after Polonius is killed. Hamlet's feelings about Laertes become very hostile, also, after the death of Ophelia. Eventually, of course, Laertes is drawn into Claudius' plot to kill Hamlet.
Hamlet's relationship with Claudius is certainly troubled from the beginning as Hamlet and Claudius regard each other with such deep suspicion. Hamlet detests Claudius for marrying Gertrude, and suspects that Claudius had killed Old Hamlet for the throne. Claudius initially attempts to befriend Hamlet, but only because he fears him. As events develop in the play, they become deadly enemies, each secretly plotting against the other.
Hamlet's relationships with Claudius and with Laertes are similar in those respects. From the beginning, both relationships are characterized by deceit. Mutual suspicions then evolve into hatred and lead to destruction.
What an interesting dynamic you have here. Hamlet seeks revenge on Claudius for the death of his father while Laertes seeks revenge on Hamlet for death of his father Polonius. Within this framework lies much to compare and contrast. You can discuss the different ways that the two avenging sons go about seeking their revenge: Laertes quite rashly while Hamlet more reluctantly. You can discuss the morality and motives of the murderers: Claudius and Hamlet. You can also discuss the reconciliation between Laertes and Hamlet at the end of the play, and the lack thereof between Claudius and Hamlet.
In your essay you should decide what you are wish to focus on. You might look at the nature of revenge and how it affects the relationship of the designated pairs. Laertes and Hamlet are in similar situations but react differently. In the end, they come to an understanding and forgive each other even though they deliver fatal wounds to each other. Claudius and Hamlet are enemies from the beginning of the play and remain so throughout. In this relationship, there is no atonement and Hamlet is clearly the victor.