What are 4-5 reasons Claudius's actions were not justified in Hamlet?

Claudius's actions are not justified for many reasons.  First, he killed his brother to take his crown. One might consider there to be some justifiable, or at least understandable, reasons for killing another person — self-defense, a crime of passion, perhaps — but killing someone for greed in such a cold-blooded way just to gain their power is not one of these potentially justifiable reasons. Thus, the murder of old King Hamlet is unjustifiable.

Second, it's not like Claudius killed a bad king or even a stranger to take his position. He killed his own brother.  As he says, it is the "primal eldest curse" (III.3.38). In referring to the biblical story of Cain killing his brother, Abel, Claudius acknowledges there is no good way to justify his murder of his brother.

Third, after Claudius killed his own brother to gain his power, he married his brother's widow. This is not justifiable because when old King Hamlet married Gertrude, she became Claudius's sister by law. Therefore, when Claudius marries Gertrude, he is (legally) marrying his own sister. Incest is not justifiable.

While it is never justifiable to marry your brother's widow, it is particularly not justifiable to marry one's brother's widow less than two months after he's died. Claudius married Gertrude so quickly that Hamlet mockingly jokes that "The funeral baked meats / Did coldly furnish forth the marriage tables" (I.2.187-188). As if it isn't bad enough to kill his brother in order to take his position, Claudius then marries his brother's wife (his own legal sister) so soon after the funeral that it hardly seems as though he (or she) has mourned the loss of King Hamlet at all. His actions seem unjustifiable in every way.