Because enotes editors can only answer one question per posting, I edited out the second quote you asked about. You may ask about that in a second posting. The reason I selected this one is because this quote is more important to the meaning of the play as a whole and more integral to an understanding of Hamlet's character.
This quote comes late in a conversation that Hamlet is having with the grave digger who is preparing Ophelia's grave (unbeknownst to Hamlet). As the grave digger is digging, Hamlet and Horatio are standing there and talking to him about what happens to the human body after death. The beginning of the conversation is somewhat light hearted -- a kind of black humor with puns and jokes. But as the conversation progresses it becomes more serious. Hamlet comes to realize that death is a great equalizer of men and that all men eventually return to dust. This is true of small people and great people such as Caesar and Alexander the Great. Through the course of the conversation, death is becoming more real to Hamlet and ultimately, more personal. This is especially true when the grave digger hands him Yorick's skull and tells Hamlet who it is. Hamlet is literally holding the skull of someone whom he spent time with and loved. Yorick was King Hamlet's court jester when Hamlet was a child and he remembers their time together. There is nothing as personal as that! Hamlet ultimately realizes that everyone is dust and to dust we all return.