How does Hamlet's father manipulate Hamlet even when he is dead?
Manipulation is a word that has negative connotations and implications, and so I wouldn't call Hamlet's father's influence on his son manipulation.
Hamlet's father, who was killed by his brother, is a tortured ghost.
Doom'd for a certain term to walk the night,
And for the day confined to fast in fires,
The relief this armored spirit can obtain is available only from his son, Prince Hamlet. The ghost speaks a simple, yet heavy request... Hamlet is asked by his farther to "Revenge his foul and most unnatural murder."
No manipulation is sought, no adjustment in thinking, no need to plan, no guile or trickery or plan of action is suggested by the ghost, for nothing of the sort is necessary.
If there is any manipulation, it is done by young Hamlet upon himself... to reorganize his thinking so that he, who is prone to thinking rather than acting, can see his way clear to act without thinking.
The fact that Prince Hamlet's father returns from the dead as a ghost to tell his son what happened to him is the basis of the entire play. Hamlet respected his father in life, and wants to do right by him in death, and that is what is central to most of the actions of Hamlet for the rest of the play.
Even before the ghost speaks he is manipulating people. He is appearing to guards and friends of Hamlet, but not speaking so as to get them to tell Hamlet about this strange thing -- a ghost. By having his friends all say they saw a ghost will make Hamlet more likely to believe what he sees and not think it is merely a hallucination caused by grief.
Once the ghost is actually talking to Hamlet he says several things to manipulate Hamlet into completing his request for vengeance. First of all he starts talking, in kind of scary generalities about where he is in the afterlife -- it is purgatory where he is "doomed for a certain term to walk the night, And for day confined to fast in fires." This creates a certain amount of sympathy in Hamlet for the horrible afterlife his father is suffering. After that is established, he kind of draws out the suspense of his message. First he mentions that he was murdered, then he tells how, and finally he delivers the fateful news, that "the serpent who did sting thy father's life now wears his crown." By building up the story this way it is manipulating Hamlet's attention and making the last note the most powerful and shocking.
The next phase of manipulation goes on for several lines when the ghost provides the more specific and gross details of his death, thus manipulating Hamlet's sympathy and anger. He also makes several negative statements about Claudius's seduction of Gertrude and how Claudius is such a lesser man than he. This feeds right into how Hamlet already feels about the gross, incestuous marriage of his mother to Claudius and how he has already expressed his displeasure at the kind of man Claudius is. In a sense, Hamlet is very prepared to hear awful things about Claudius -- he thought those very things even before he learned that Claudius murdered his father.
The ghost also manipluates Hamlet by drawing a sympathetic picture of Gertrude so that Hamlet won't try to punish her for her action in marrying Claudius. Claudius is the only one that the ghost wants revenge on. He tells Hamlet to leave his mother to heaven's judgment.
His final manipulation is to the command to "remember me." He wants Hamlet to have at the forefront of his thoughts the father that Hamlet used to be and the facts that he learned today. Together, all of that should keep Hamlet highly motivated to avenge his father's death by taking care of Claudius once and for all.