What does Hamlet mean by the quote from Act V, Scene 1, "Let Hercules himself do what he may..."?
Hear you, sir; What is the reason that you use me thus? I lov’d you ever: but it is no matter; Let Hercules himself do what he may, The cat will mew, and dog will have his day.
The context for this quote is Ophelia's funeral. Laertes has, in his grief, jumped into Ophelia's grave, and Hamlet follows him. Laertes attacks him, at which point Hamlet asserts that his love for Ophelia is far greater than that of Laertes. After a lengthy tirade, he asks Laertes why he had attacked him, because he had always been his friend. The last lines are saying, essentially, "what will be, will be." Even someone as strong as Hercules can't stop the course of events. This suggests that Hamlet believes fate is at work in the series of events that have taken place. It also seems to both foreshadow the violent ending while it sends the message that even his grief at the death of his beloved Ophelia (and the anger of her brother) has not put him off his sworn objective of avenging his father's murder.
“Let Hercules himself do what he may, The cat will mew and every dog will have his day.” Hamlet directs this statement to Laertes to express his feelings about the death of Ophelia. Hamlet had returned from England, where King Claudius had arranged for his death through a letter, which Hamlet discovered and altered. Upon his return, he stumbled upon Ophelia’s burial ceremony during which Laertes jumps in the grave, expressing his grief at the loss of his sister. Hamlet also jumps in, claiming he loved Ophelia more and stating that Laertes was just being overly dramatic. They fight and the King orders for their separation. It was at that moment after the fight that Hamlet made the statement which meant, like Hercules, Laertes may be strong but his actions would not bring Ophelia back. In the phrase, Hamlet also suggests that in the end he will have his way regardless of Ophelia’s death, since it was the price he had to pay to avenge his father and seek justice for his family.