In the third solioquy, explain the central problem Hamlet faces in deciding whether or not to murder Claudius?
In the soliloquy "O what a rogue and peasant slave am I" Hamlet berates himself for not having killed Claudius yet. He piles insult after insult upon himself, the gist of which is that he is a coward. He has just seen the Player recite a set-piece from a play about the Trojans at the end of the Trojan War. The Player cries as he talks of the Trojan Queen Hecuba. The tears come from passion, but the man himself (i.e., the actor) has used his craft to present a character in a play crying on stage.
He thinks he knows that Claudius is guilty because the Ghost told him so. But does he really? There are many reasons for doubting the veracity of a ghost in Shakespeare's time, but you don't have to go outside of the play to find them. Hamlet and Horatio tell us these problems in a number of speeches. So an unstated (at least in this speech) relates to the veracity of the Ghost. That's why Hamlet wants to do "The Mousetrap" before the King--so he can tell by Claudius's reaction whither he really killed Hamlet's father.
So there are two major problems expressed here. 1) Hamlet's perceived cowardice and 2) Hamlet's need to have absolute truth about Claudius killing Hamlet Sr.