What does the ghost request of Horatio and Marcellus in Hamlet?

While the ghost of King Hamlet appears to Horatio and Marcellus, two officers who are on guard, it does not actually speak to either of them. The next evening the ghost speaks to Prince Hamlet, asking him to avenge his death because he was murdered.

After these officers identify the ghost as that of King Hamlet, dressed in the armor which he wore when he was slain, they report to Hamlet what they have seen. Prince Hamlet asks Horatio if he did not speak to this ghost. Horatio replies, 

My lord, I did,
But answer made it none. Yet once methought
It lifted up its head and did address
Itself to motion, like as it would speak.
But even then the morning cock crew loud,
And at the sound it shrunk in haste away
And vanished from our sight. (1.2.214-220)

When Horatio was with Marcellus and the ghost acted as though it would speak but the rooster crowed, it hurried away because, according to legend, the rooster awakens the god of day. This threat of the approach of dawn makes all wandering ghosts hurry to the places where they hide. And, the ghost of King Hamlet did what other wandering spirits do--he hastened to his hiding place.

Since King Hamlet's ghost has not spoken to anyone, Hamlet instructs Horatio and Marcellus that he will meet them on the guards' platform between eleven and midnight. At that time he will attempt to speak to this ghost of his father if he will reappear.

On the next night, Hamlet arrives as he has promised. The ghost does reappear, but he draws Hamlet away from the others. Nevertheless, Marcellus is determined to follow as he suspects "Something is rotten in Denmark" (1.4.90). After the ghost has led Hamlet away from the others, Hamlet tells the ghost he will go no farther. "Speak, I am bound to hear" (1.5.7). The ghost identifies himself as the spirit of King Hamlet, telling him he must hurry because he is under obligation to return to the fires of purgatory to atone for his sins until he can go to heaven. The ghost then asks Hamlet to "Revenge his foul and most unnatural murder" (1.5.25).