King Hamlet's story

Henry Fuseli, Horatio, Hamlet, and the Ghost, 1798

In Act I, scene v of Hamlet, young Hamlet encounters the ghost of his father, a spirit at unrest because of the nature behind his death. During this scene, his father opens up to Hamlet the true reason for his death, revealing the evil deeds of Claudius and Gertrude, the new king and his newlywed queen. 

King Hamlet's passionate anger towards his brother can be seen when he calls him "that incestuous, that adulterate beast" (I.v.49). This is similar to when young Hamlet compares his mother to a beast when referring to her marriage to Claudius only two months after King Hamlet's death. He mentions how "a beast that wants discourse of reason / Would have mourned longer" (I.ii.155), showing how he sees his mother as less than a beast lacking the ability to reason. 

Hamlet's father continues to say how his wife was won over by Claudius' gift to seduce, referring to his wife as "seeming-virtuous" (I.v.53). The use of "seeming" portrays the queen as only having the appearance of being of virtues. This is similar to when young Hamlet speaks about seeming versus being, that seeming is simply "actions that a man might play" (I.ii.87), showing that Gertrude is simply acting as virtuous as necessary to uphold her reputation. His father is implying that it isn't only Claudius who fooled the state but also Gertrude, whose "lewdness court [virtue] in a shape of heaven" (I.v.61). King Hamlet speaks about how his love for his wife was of dignity, while her love and gifts "were poor / To those of [his]" (I.v.58-59). 

After his death, King Hamlet appears to be aggravated by the fact that he died without any final rites, "unhousseled, disappointed, unaneled" (I.v.84). He mentions how he is laid to rest with imperfections on his head, implying that he still has regrets and guilts he's unable to overcome because of his death. This could imply that he's currently not in hell but rather purgatory, as he makes mentions of how he is "confined to fast in fires" (I.v.16) until his crimes have been "burnt and purged away" (I.v.18). His death before he could repent for his misdeeds would force him into purgatory, where he would have to atone for his sins until he be allowed to enter heaven. 

Near the end of the ghost's story, young Hamlet is told to not let the "royal bed of Denmark be / A couch for luxury and damnèd incest" (I.v.89-90). In addition to pleas before about obtaining revenge for his murder, the King is basically telling Hamlet to act out against his uncle, perhaps even to the lengths of murdering him. After all, achieving this revenge would be through killing the killer. King Hamlet tells Hamlet to leave his mother to alone, to "leave her to heaven" (I.v.93), where there she will have to repent for the misdeeds she's committed. 

It doesn't seem as though King Hamlet realizes the extent of the request he asks of young Hamlet. By asking that he seek revenge for his murder, King Hamlet is upsetting the state of their kingdom. Although the state is already unrest, seeing as the king was murdered by his brother, his request is inevitably going to drag the state even further into unrest, by having his son kill the new king. Though his desires are understandable, he isn't thinking through what he's asking young Hamlet to do, which is basically to commit treason as Claudius had done.