It is actually the ghost who utters the words “Remember me” in Shakespeare's Hamlet. Let's look at the context of those words.

It is act 1, scene 5, and Hamlet is meeting his father's ghost. The ghost is “Doom'd for a certain term to walk the night, / And for the day confined to fast in fires” until his sins are “burnt and purged away.” But he has a revelation and a command for his son. The ghost died as a result of murder, and the murderer is his own brother, Claudius, now king in his place and husband of Hamlet's mother, Queen Gertrude.

The ghost tells Hamlet how he was killed. He also tells his son to avenge his death. Yet Hamlet is not to harm his mother or even think ill of her. He is to let God judge her for her acts.

As the ghost leaves, he says, “Adieu, adieu! Hamlet, remember me.” And Hamlet does remember everything the ghost has told him. He struggles with the command for revenge, yet in the end, he takes it, only to lose his own life in the process.