Immediately after Hamlet speaks with the ghost, he tells Horatio, Marcellus, and Bernardo, "It is an honest ghost, that let me tell you" (I.v.152). It's not that Hamlet doesn't believe his father's ghost, but if he went into Claudius and told him that he knew from a ghost that he is a murderer, he wouldn't have much credit with that story and be considered crazy. Further, his story would not hold up in court if he said that he learned all of his intelligence by way of a ghost. Later, after Hamlet meets the players and is alone, he realizes that he can have the actors play out the scene of how his father was killed to see how Claudius and Gertrude act when it is performed. This is a much better and indirect way of discovering Claudius's guilt. Hamlet does question if he is just being a coward with the whole thing, but continues with the plan of the play anyway.
"Out of my weakness and my melancholy,/As he is very potent with such spirits,/ Abuses me to damn me. I'll have grounds/ More relative than this. The play's the thing/ Wherein I'll catch the conscience of the King" (II.ii.596-600).
However, it's really not Hamlet's doubts about the ghost's honesty or ambition that hinders his inaction to kill Claudius, but his inability to justify himself becoming a murderer and then finding the right opportunity to carry out the task. Hamlet is young and over-burdened with the faults and sins of his parents. It is a lot to ask for him to kill Claudius and leave his mother to the judgments of God. Hamlet is almost more hurt about his mother's involvement in the murder as he was surprised that his uncle could kill his father. Whether it is cowardice or reason that motivates Hamlet to have the players act out the murder scene isn't really clear, but the end result is certainly effective enough to keep Hamlet on the path of revenge.