How important is the ghost who appears in Act I of Hamlet?

The ghost of King Hamlet who appears in Act I is intrinsic to the plot and the character development of Prince Hamlet.

It is the ghost of his father who alerts Hamlet to the murderous character of his uncle Claudius, as well as "the something rotten in Denmark"(1.4.99). Stirred greatly by his encounter with the ghost, Hamlet vows revenge; however, he moves at first only from one emotional soliloquy to another. For, his melancholic personality mulls over what his father's spirit has told him, and Hamlet feels he must prove that Claudius is, indeed, the murderer of his brother. Then, when the Dumb Show scripted by Hamlet proves that Claudius suffers guilt, Hamlet follows his father's murderer. But, as he sees Claudius in prayer, Hamlet hesitates because he does not want to make a martyr of his uncle:

Now might I do it pat, now 'a is a-praying,
And now I'll do't--and so 'a goes to heaven,
And so I am revenged....
Why, this is hire and salary, not revenge. (3.3.73-79)

Again, Hamlet retreats into his deep melancholy initiated by the ghost and his knowledge of the rottenness in Denmark. But, in Act III, Scene 4, when Gertrude summons Hamlet, he storms into her chambers, demanding, "Now, mother what's the matter?" (3.4.7) and she tells him he has offended his father, meaning Claudius. Hamlet retorts that it is she who has offended his father, meaning King Hamlet. Moreover, it is during this interchange that Hamlet stabs Polonius who is behind the arras, thinking that Claudius is hidden there. 
Angered further by what has occurred, Hamlet denounces his mother for her incestuous marriage, and he rails against Claudius. Suddenly, the ghost of King Hamlet intervenes, reminding Hamlet of his true purpose. This ghost points to the "amazement on thy mother sits" (3.4.114), further directing Hamlet to not chide his mother because she obviously does not know what Claudius has done. Because the ghost of his father then asks him to intercede on his behalf with Gertrude, Hamlet tries to get his mother to see this spirit so she will know what has happened. However, she is unable to do so, and the ghost disappears.

Again, although he has been reminded by the ghost of his purpose, Hamlet retreats into his deep melancholy. And, it is not until he witnesses the courage of young Fortinbras who is willing to die, "[E]ven for an eggshell"  (4.4.53), in order to avenge his own father that Hamlet finally responds to the demands of his father's spirit. Declaring, "This is I/Hamlet the Dane" (5.1.227-228), Hamlet takes action as he duels Laertes in the final act and kills Claudius, ordering him,

Here, thou incestuous, mur'drous, damned Dane,
Drink off this potion....(5.2.304-305).

Prince Hamlet achieves the revenge sought by his father's ghost, but it is tragically attained at the cost of both his mother's life and his own. And, so the ghost has effected both vengeance and death, dominating both the characterization and action of the drama.