The apparent normality of the play is suggested by Claudius when he says to Hamlet that all men lose fathers and that the death of Hamlet's father was a natural event. However, when Hamlet sees the ghost of his father, the ghost tells him that he was murdered by his brother. The fact that a ghost would even be seen and that Hamlet would even speak to it also disturbs apparent normality. That's why the first line of the play, "Who's there?" is so telling. In this play, we really must learn the difference between normality and abnormality. We have no idea who is really "there".
Shakespeare enjoys employing ghosts and witches and fairies however. Fairytale elements. So what is to say what is normal and what isn't in their society?
What is abnormal is the unnatural murder of a king. The king is the highest authority. The fact that Hamlet's father tells Hamlet that he was murdered means something really is rotten in the state of Denmark.
Don't take me for the last word though. You know. ;)