Final answer:
Hamlet is offended by Claudius's revelry and drinking, which he sees as a betrayal to the memory of his deceased father, King Hamlet.
Step-by-step explanation:
While Hamlet is out with Horatio and the guards looking for the ghost of King Hamlet, King Claudius is indulging in the pleasures of the court, such as drinking and revelry. This is offensive to Hamlet because it shows a stark contrast between Claudius's behavior and the mournful respect Hamlet believes is due to his deceased father, King Hamlet.
While Hamlet is out with Horatio and the guards looking for the ghost of King Hamlet, King Claudius is busy attending a party with his courtiers. Hamlet finds this offensive because he believes that King Claudius is not grieving the death of his father and instead indulging in worldly pleasures.
In Act I, Scene 2 of Shakespeare's Hamlet, Hamlet compares his father to the god Hyperion and his uncle Claudius to a satyr, highlighting Claudius's inferior nature and lustful, hedonistic behavior. Hamlet finds it offensive that Claudius has married his mother so quickly after his father's death and is behaving in such an undignified manner, which he views as a betrayal to his father's memory.