What are Claudius's sins?

After Claudius's reaction at the end of The Murder of Gonzago, the reader should have no doubt that Claudius is guilty of his brother's murder. Just in case the reader wants further proof, Claudius speaks a monologue on his own, saying, "O, my offense is rank, it smells to heaven." Then he uses exact wording by saying the following:
But, O, what form of prayer / Can serve my turn? "Forgive me my foul murder"? / That cannot be, since I am still possessed / Of those effects for which I did the murder, / My crown, mine own ambition, and my queen.
Claudius actually admits to the murder. In fact, Claudius actually names his own sins when he says, "My crown, mine own ambition, and my queen." Claudius's brother (Hamlet's father) had the position of power, the kingship. Claudius demonstrates envy because he is willing to do anything to get into that position of power. As a result of his selfish desires, Claudius is definitely guilty of greed. Finally, due to Claudius' additional desire for Gertrude, Claudius is also guilty of lust. Why are these sins of envy, greed, and lust important? They are three of what the Roman Catholic Church called "the seven deadly sins." At the time, people were taught that all sins stemmed from these seven as the "origin" of all sin. In a way, this shows that Claudius is in no way a moral and noble character.