In the context of this play, when kings held absolute power and there was no independent justice system, can you think of any way to deal with Claudius other than to kill him?

It's not clear in the play that Claudius holds absolute power. After all, he has to resort to underhanded means to try to get rid of his feared rival, the young Hamlet. He sends him off to England with secret letters in the hands of courtiers asking the English to kill Hamlet, because Claudius doesn't dare do this in Denmark. And he tries, successfully, to persuade Laertes to poison the tip of his rapier before his sword fight with Hamlet, hoping to kill him in that devious way.
If Claudius had had the absolute power to clamp Hamlet into prison on trumped-up charges and then have him executed, he would have. But Claudius's lack of absolute power is where Hamlet has his best chance.
Claudius tries to get Hamlet to England and killed there, far away, because he knows Hamlet, who probably should have been named king, is popular with the common people and has a legitimate claim to the throne. Hamlet could have appealed directly to the people to rise up and support him against Claudius—and probably could have been successful, driving Claudius to flee into exile.