What differences are brought out between the characters of Hamlet and Claudius in Act one, scene 4?
In this scene, we learn about Claudius' character from the conversation Hamlet has with his friend, Horatio. Horatio has brought Hamlet along with him in the dead of night in order that his friend might see the apparition of his father, the king, whom his Uncle Claudius had murdered.
Before the ghost appears, the two friends hear the sounds of trumpets and merriment:
The king doth wake tonight and takes his rouse,Keeps wassail, and the swaggering up-spring reels;And as he drains his draughts of Renish down,The kettle-drum and trumpet thus bray out
The triumph of his pledge (1.4.9-13)
Hamlet explains that the king is partying rather than sleeping ("wassail" is an alcoholic drink) and "reels" are a wild German dance.) In the last line here, the king has drained his cup after making a toast.
Cladius' behavior stands in stark contrast to the somber and just King Hamlet. Claudius' behavior is even more appalling when he has come to the throne so treacherously.