Hamlet's first important soliloquy, "O that this too sullied flesh would melt...", ushers in what later becomes a central idea in the play - Hamlet's thoughts of suicide. The quote illustrates the intense grief that Hamlet is experiencing because of not only the death of his father, but also the almost immediate marriage of his mother to his uncle, so soon after his father's passing. Through this speech, Hamlet's chracter is revealed because of the obvious exasperation with which he regards his life. He feel as if he has nowhere to turn - neither his religion nor his family can provide any comfort to him.
Hamlet desires to wipe himself clean of all the impurities than he can see around him by suicide, however he realizes that by killing himself, he woudl actually be defiling his name even more. This serves to provide an exposition about not only Hamlet's character but also his possible actions later on.
The diction choiuces such as "sullied flesh" rather than "desecrated skin" help to give the reader more of an insight as to what kind of a character Hamlet is. He is passionate and clearly doesn't know what to do. These themes of suicide, emptiness, and degradation also serve not only to advance the plot but also to provide a more intense character analysis of Hamlet himself.