What delicate issues are at the very top of Claudius's agenda for the meeting of his court in Hamlet?
Just to clarify some things:
By the time of the speech before the court (Act 1 scene 2), Claudius and Gertrude are already married. Indeed, Claudius generally thanks those gathered for previously offering their advice about the marriage.
The next bit of business is of note in that it involves young Fortinbras, the son of the elder Forintbras who was killed by Prince Hamlet's father. It also involves an uncle, the elder Fortinbra's brother who is sick and knows little about young Fortinbra's plan to reclaim land lost by Norway to the dead King Hamlet. Claudius says that he plans to send Cornelius and Voltimand to get the old uncle to stop young Fortinbras in his foolish venture.
Next, the King brings up the subject of Laertes who wants to go back to school. After asking Laertes' father, Polonius, if it's OK with him, Claudius grants Laertes his wish.
The same wish, to go back to school, is not granted to Hamlet. In this last bit of business before the court, the King is adamant: Hamlet is to stay for a while in Denmark. The King wants to keep an eye on Hamlet, but he certainly does not share this reason, nor his wariness, with the rest of those assembled.
I assume you are talking about the meeting between Claudius and some members of his court in Act I, Scene 2.
In that meeting, perhaps the most important thing that is going on is the relationship between Denmark and Norway. Claudius is worried about the behavior of Fortinbras, Prince of Norway. So he sends Voltemand and Cornelius to the King of Norway (who is Fortinbras's uncle) asking him to restrain his nephew. Fortinbras is angry because Hamlet's father killed his father and he wants revenge on Denmark.
Other than that, the main isssue is telling the court that he has married Gertrude. He and Gertrude also try to get Hamlet to stop mourning his father.
Claudius announces that he intends to hurry and marry the Queen. He states that he is doing it in haste because the Danes are coming. Claudius, the brother of the now dead king, has gained the rights of the throne and the power. Claudius knows that Hamlet is the kings son and the follow up heir to the throne. He intends to make Hamlet feel foolish and minute. He also wants his kingdom to stop mourning the king even though it has only been a couple of months. He is manipulative and deceptive. Act I, Scene 2.