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The play The Murder of Gonzago had a plot quite similar to the account the Ghost had given of the murder of Hamlet Sr. With a few tweaks, it was so similar that it reminded Claudius of his murder which was what Hamlet intended.
The soldiers want to tell about the ghost to scare you.
The play had been chosen and tweaked by Hamlet to resemble as closely as possible the murder of Hamlet Sr. by Claudius according to the ghost. Since the ghost was in fact telling the truth, Claudius did not want to have his crime played out in front of him. He had a number of reasons: first, as his subsequent soliloquy shows, he feels guilt about the murder and wishes he could be free of that guilt. Also, some of the audience might have made the connection and have suspected him of killing Hamlet Sr. who everyone thought had died of snakebite.
A number of things, obviously. The Ghost is still bellowing "Swear!" from under the stage. And we can assume that Claudius is still partying as he was earlier in the previous scene. What else is happening, we don't know.
They tell hamlet.
First, Hamlet is popular and Claudius wouldn't want to start a popular uprising. Second, he doesn't want to cause pain to Gertrude, who he genuinely loves and does not want to hurt.
Hamlet.
Hamlet was causing problems for Claudius. His antics had disturbed the court and the kingdom. His killing of Polonius had deprived Claudius of his counsellor and had put him in an awkward position, since a murderer ought to be tried and put to death yet that would be politically unwise. Finally, Hamlet was on to him. Claudius had a pretty good idea that Hamlet knew that he had killed Hamlet Sr. and how it was committed. All this meant that Claudius was worrying about Hamlet instead of running the kingdom, which he had been doing quite well before all this. Claudius says, "like the hectic in my blood he rages."--it is becoming an obsession.
Claudius wants Hamlet dead. He wants his own hands clean, though. His main priority right then is appeasing Laertes, but that aligns nicely.
Well, the basic reason is that they're not in the Chapel. The setting is Claudius's private room, the King's Room. That's stated in the play dialogue. Guildenstern says it: "(the king) is in his retirement..." It's reference to the place Claudius retires for the night, his private quarters. When Claudius left the 'Mousetrap' play, he went to his own room. Claudius prays during the scene, and apparently the idea of praying has led some people to think the scene is in the Chapel, however, the actual play dialogue tells us otherwise. People don't have to be in a chapel to pray, and many people pray in their own rooms, of course, which is what Claudius does. Hamlet is discouraged from killing Claudius because he finds Claudius praying. Hamlet is afraid that if he kills Claudius at prayer, Claudius's soul will go straight to Heaven. However, the Ghost said, or very strongly implied, that the soul of Hamlet's father was in Purgatory. Hamlet doesn't see it as fair, that he could send Claudius to Heaven, when his own father is in Purgatory. Hamlet decides to wait for another opportunity later, when there's less chance of sending Claudius's soul to Heaven.
praying
Hamlet