Lines 333-334: Hamlet is saying that he wishes his body would dissolve into a puddle of its own accord. In other words, he is saying he doesn't want to exist any more.
Lines 335-336: He also wishes that it wasn't against the laws of God to commit suicide.
337-338: He is saying that all the joy has gone out of life and its pleasures.
339-341: Hamlet likens life to a garden that has been allowed to run wild and grow gross and disgusting things in it as a result of a lack of tending.
342: The person he is speaking of (his father, King Hamlet) has been dead for less than two months.
343-346: Hamlet says his father is a great king and compares him to Hyperion (one of the mythological Titans, a god of light and wisdom) and his uncle Claudius to a satyr (a mythical part-human-part-animal monster with a constant, exaggerated erection). He goes on to say this his father was so loving to his mother that he would stop the very winds from blowing too hard against her face.
347-349: Hamlet describes the way his mother used to dote on his father as if all of the time she spent with him constantly increased her desire for more. He ends line 349 with the acknowledgement that "yet, within a month..." we are left to assume he means that even within a month she was considering remarriage.
350: Hamlet refuses to finish the previous thought and states that women are the embodiment of weakness.
351-352: He describes how it has only been a month and his mother's brand new shoes that she wore to walk in his father's funeral procession are not even broken in yet.
353: He likens his mother's behavior at the funeral to Niobe, a figure from Greek mythology who wept for nine days and nights when all her children were slain by the gods. (And implies that even still, she didn't stay faithful to his father's memory for long.)
354-359: Hamlet claims that even a brainless beast would have mourned a loved one longer. He discusses how his mother not only didn't mourn for long, but she married her dead husband's own brother. He also states that Claudius and King Hamlet were as different from each other as Hamlet himself is from Hercules. The reader is meant to understand that serious, scholarly, melancholy Hamlet is very different from the mythological hero, Hercules, a man of action and strength (and not really one of intelligence).
360-361: He complains that she married with "wicked speed" and got into bed with her brother-in-law before the salt of her tears for King Hamlet had even dried.
362-363: Hamlet thinks things will turn out badly, but he knows he can't protest openly.
Summary: Act I, scene ii
The morning after Horatio and the guardsmen see the ghost, King Claudius gives a speech to his courtiers, explaining his recent marriage to Gertrude, his brother’s widow and the mother of Prince Hamlet. Claudius says that he mourns his brother but has chosen to balance Denmark’s mourning with the delight of his marriage. He mentions that young Fortinbras has written to him, rashly demanding the surrender of the lands King Hamlet won from Fortinbras’s father, and dispatches Cornelius and Voltimand with a message for the King of Norway, Fortinbras’s elderly uncle.
His speech concluded, Claudius turns to Laertes, the son of the Lord Chamberlain, Polonius. Laertes expresses his desire to return to France, where he was staying before his return to Denmark for Claudius’s coronation. Polonius gives his son permission, and Claudius jovially grants Laertes his consent as well.
Turning to Prince Hamlet, Claudius asks why “the clouds still hang” upon him, as Hamlet is still wearing black mourning clothes (I.ii.66). Gertrude urges him to cast off his “nightly colour,” but he replies bitterly that his inner sorrow is so great that his dour appearance is merely a poor mirror of it (I.ii.68). Affecting a tone of fatherly advice, Claudius declares that all fathers die, and all sons must lose their fathers. When a son loses a father, he is duty-bound to mourn, but to mourn for too long is unmanly and inappropriate. Claudius urges Hamlet to think of him as a father, reminding the prince that he stands in line to succeed to the throne upon Claudius’s death.
With this in mind, Claudius says that he does not wish for Hamlet to return to school at Wittenberg (where he had been studying before his father’s death), as Hamlet has asked to do. Gertrude echoes her husband, professing a desire for Hamlet to remain close to her. Hamlet stiffly agrees to obey her. Claudius claims to be so pleased by Hamlet’s decision to stay that he will celebrate with festivities and cannon fire, an old custom called “the king’s rouse.” Ordering Gertrude to follow him, he escorts her from the room, and the court follows.
Alone, Hamlet exclaims that he wishes he could die, that he could evaporate and cease to exist. He wishes bitterly that God had not made suicide a sin. Anguished, he laments his father’s death and his mother’s hasty marriage to his uncle. He remembers how deeply in love his parents seemed, and he curses the thought that now, not yet two month after his father’s death, his mother has married his father’s far inferior brother.