I agree: One would assume that the Ghost, whom other people have already seen, who looks exactly like Hamlet's father in armor, who sounds like him, who knows all about the specifics of the murder, and who will only speak to Hamlet, such a Ghost would not be doubted. I don't believe in ghosts, but I'd believe that one. And further, right there in Act 1, Scene 5, immediately after seeing the Ghost, Hamlet says to his friends:
Touching this vision here,
It is an honest ghost, that let me tell you.
Later, though, just before the play-within-a-play, Hamlet tells Horatio to watch Claudius' reactions (Act 3, Scene 2) for any sign of uneasiness:
If his occulted guilt
Do not itself unkennel in one speech,
It is a damned ghost that we have seen,
And my imaginations are as foul
As Vulcan's stithy.
Does this mean that Hamlet, at some point, has had second thoughts? Maybe. Perhaps he's trying to be ultra scientific, trying to make double sure? or maybe he's just putting things off a little so that he can really irritate, anger and then ultimately catch the conscience of the King.
This seems his likely motivation, for when, after the pay, he sees the King trying to pray, he does not carry out the revenge. Hamlet doesn't want just any revenge: he wants it sweet. Not just a simple killing but one worthy of the dear father he has lost.
In my opinion, Hamlet really doesn't need proof. What he needs is determination. However, if we go on the assumption that he does need it, I would sayt that Hamlet needs proof simply because he is so indecisive. If he had a stronger, more aggressive personality, he probably would have killed Claudius without proof.
You would think that Hamlet has (early on in the play) proof enough that Claudius killed Hamlet's father. This is because his father's ghost tells him so. Hamlet does not express any doubt that this is true.
But he still doesn't kill Claudius. Instead, he does things like making the actors do a play that will force Claudius to reveal what he has done -- thus giving Hamlet further proof. Of course, he still doesn't kill Claudius right away...
Anyway, I think he is looking for more proof simply as a way to put off doing something that he thinks he should do, but doesn't really want to do.