Line of kings: Hamlet senior to Claudius?

Usually, when a king dies, his throne is passed down to his eldest son. When Hamlet senior died, the crown was not passed to his son, but to Claudius. I think that it is clear that Claudius married the queen because he wanted to be king. This leads me to really not like Claudius. This also seems to have major repercussions on our Hammy junior. Our Hamlet seems to have tons of tension with Claudius, and although they don't directly argue and yell at each other, they discuss with passion and, frankly, hate. Claudius fires hurtful words to assert that our young Hamlet is not manly. He also does not want him to go back to school for some reason. Hamlet junior is depressed, and does not know what to do.

On a separate, and less serious note, it is very difficult for me to tell the difference between all the characters. I feel like I hear a bunch of names, but the few that really stand out as characters I really remember are both Hamlets, the queen, Claudius, and sort of Laertes and Polonius.