murder, albeitPolonius mistakenly, within a few minutes of the mousetrap scene.  So the issue is not a matter of believing the ghost, because Hamlet does at this point.  His greater problem is doing away with Claudius swiftly and secretly. When Hamlet overhears Claudius' confession, he is reaffirmed that the ghost is a honest one.  It would be easy to assume that Hamlet does not kill Claudius at confession because there is lingering doubt, but in fact that is not the case. I'll explain briefly:

The reason that Hamlet does not murder Claudius at confession is because in the tradition of the Catholic church, Claudius would have gained passage into heaven because he has begun the process of confession and repentance.  In other words, Claudius would go to heaven if murdered at confession.  King Hamlet, meanwhile, who was not given a chance to confess and repent before his death, is in purgatory.  Hamlet elects not to murder Claudius at confession because he'd rather catch the new King in a sinful act and murder him there, before Claudius has a chance to repent.  That way his soul will also go to purgatory.

Prince Hamlet is depicted as an insecure, conscientious individual who is very religious and fears that seeking revenge on King Claudius could potentially damn his soul. It is important to note that many of Hamlet's significant decisions are associated with his religious beliefs, which is why he refrains from committing suicide. In act 1, scene 5, Hamlet speaks to his father's ghost and learns of Claudius's treachery. The Ghost then instructs Hamlet to avenge his death by murdering the king, which is something Hamlet vows to do. Later on, Hamlet becomes hesitant to take action and begins to doubt the Ghost's message. As a moral, conscientious person, Prince Hamlet does not want to risk damning his soul by unjustly committing regicide and is also overwhelmed with complicated emotions, which is why he begins to question the validity of the Ghost's message.

In his soliloquy toward the end of act 2, scene 2, Prince Hamlet mentions,

The spirit that I have seen
May be a devil, and the devil hath power T'assume a pleasing shape, yea, and perhaps,
Out of my weakness and my melancholy—As he is very potent with such spirits —Abuses me to damn me. (Shakespeare, 2.2.586–591).

Hamlet's comments depict his cautious disposition and discerning nature. Hamlet recognizes that he is in a dangerous environment where no one can be trusted. He has recently learned of his uncle's treachery, witnessed his mother's immediate marriage, spoken to his father's disturbed spirit, and has been made aware that Claudius's agents are spying on him. Given the dangerous, uncertain atmosphere of Denmark and the king's deceiving nature, one can argue that Hamlet is predisposed to act cautious and question everything, which includes the Ghost's message. Overall, Prince Hamlet questions the validity of the Ghost's message as a natural response to the strange, threatening climate surrounding Elsinore and out of fear that he will damn his soul by unjustly committing regicide.

The 17th century was very much an age of doubt. The scientific revolution called into question many of the old certainties in relation to our understanding of the cosmos and its workings. And in the figure of Hamlet—the brooding, introspective student prince—we have the ideal embodiment of this increasingly skeptical era.

Hamlet doesn't actually doubt the Ghost's testimony; in fact, it confirms his worst suspicions about Claudius. But being such a thoughtful soul, he's not about to dash off straight away and run through his wicked uncle with a sword. He's going to exact a terrible revenge on Claudius, but in his own way and at his own pace. What the Ghost said is probably true, but Hamlet wants to know for sure.

So he embarks upon an elaborate charade which involves feigned madness and the staging of a play called The Murder of Gonzago. "The play's the thing" for Hamlet; he's going to rewrite it to depict the precise details of his father's murder. This will give him the opportunity to probe Claudius's reaction. If he's guilty of this wicked crime, then he'll squirm uncomfortably in his seat, and then Hamlet will know with absolute certainty that the Ghost was telling the truth.

For Hamlet, seeing is believing. Once he's seen the look on Claudius's face as the murder scene plays out, then every last trace of doubt will have been expelled. For Hamlet, as with any self-respecting intellectual of the early modern era, doubt is the catalyst for sustained rational inquiry, the gathering of empirical evidence that points toward the truth of the matter.

Hamlet does not doubt that the ghost he meets in Act 1, Scene 5 is truly the ghost of his dead father. Later, however, he begins to have doubts. In his soliloquy at the end of Act 2, Scene 2, he explains why.

The spirit that I have seen
May be a devil; and the devil hath power
T' assume a pleasing shape; yea, and perhaps
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.

The arrival of the players has given him an idea. He will have them put on a play which resembles what the ghost has told him. A man murders a sleeping king and seduces the king's wife. The play is titled The Murder of Gonzago. Hamlet tells himself in his soliloquy that he will observe Claudius closely in order to see how he reacts. As Hamlet says:

I'll observe his looks;
I'll tent him to the quick. If he but blench,
I know my course.

Later, in Act 3, Scene 2, he asks Horatio to help him watch Claudius during the play. By this time Hamlet has taken Horatio into his confidence and told him what he heard from the ghost. Horatio agrees to Hamlet's request. The audience will also be watching Claudius very closely after all this preparation. Shakespeare creates a strange stage effect with this play within a play. The audience is watching a play in which an audience is watching another play. The fact that the audience in the play is watching a play makes the real play seem more like reality. Claudius does not seem like a character in a play at this point but just another member of the audience. Shakespeare's audience is led to believe they are watching someone in the audience and not an actor on the stage. Claudius makes a spectacle of himself when he sees the character Lucianus pour poison into his sleeping uncle's ears. Claudius calls for lights and flees ignominiously. Hamlet is delighted that his trick has worked. Horatio confirms what Hamlet now believes.

HAMLET
O good Horatio, I'll take the ghost's word for a thousand pound. Didst perceive?

HORATIO
Very well, my lord.

From now on Hamlet will be determined to assassinate Claudius, and Hamlet now has an ally in his friend Horatio, who is also convinced that Claudius murdered Hamlet's father the former king. Hamlet calls him "O good Horatio" because he is so happy to have someone on his side. Hamlet has been all alone up to this point, surrounded by spies and false friends. There had been no one he could trust, including Ophelia. Now he can trust himself, trust Horatio, and trust his father's ghost. This is one of the places in the play where Hamlet shows an evolving character change. He now has more self-confidence. He trusts his own intuition.

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.

It is not unreasonable that Hamlet might be having second thoughts about his encounter with the apparition who claimed to be his father. Hamlet was in a highly emotionally charged state. He had never seen a ghost before. He might not necessarily believe in ghosts. His behavior is erratic when he rejoins Horatio, Marcellus and Bernardo. And the information the ghost imparted was totally astonishing. When Hamlet has regained his composure he suggests to himself that the spirit might have been the devil.

The spirit that I have seen
May be a devil; and the devil hath power
T' assume a pleasing shape; yea, and perhaps
Out of my weakness and my melancholy,
As he is very potent with such spirits,
Abuses me to damn me.      (Act II, Scene 2)

If it really were the devil attempting to induce Hamlet to commit a mortal sin, there is no question that such a powerful being would be able to assume any shape he wanted and would be able to exert a potent influence on any mere mortal. 

It is characteristic of Hamlet to have second thoughts, third thoughts, and any number of thoughts. His tragic flaw, as has been frequently pointed out, most notably by Samuel Taylor Coleridge, is that he thinks too much. He is frequently second-guessing himself. An excellent example is the way he stops himself from killing Claudius when he finds him alone at prayer. Hamlet says:

Now might I do it pat, now he is praying;
And now I'll do't. And so he goes to heaven,
And so am I revenged. That would be scann'd.
A villain kills my father; and for that,
I, his sole son, do this same villain send
To heaven.                   (Act III, Scene 3)

By "That would be scann'd," Hamlet means the question needs deep consideration. This is just the sort of thing he delights in, because he is so adept at doing it. He has developed a habit of "scanning" everything. His years at Wittenberg have given him the mental tools for doing it with precision.

Hamlet's doubts about the identity of the spirit he met on the battlement and about the information he received there lead him to staging a play titled The Murder of Gonzago which he hopes will force Claudius to betray his guilt. The trick is successful beyond Hamlet's highest expectations. Claudius is horrified when he sees the player doing exactly what he did himself, pouring poison in a sleeping man's ear. Clausius does not merely betray his guilt in his facial expressions; he creates a big scene by disrupting the play-within-a-play, calling for lights, and fleeing from the room. After this, Hamlet is convinced and decisive. It is true that he does not kill Claudius when he finds him at prayer after the play-within-a-play is disrupted; but what is important is that Hamlet has definitely decided that Claudius is going to die. He puts up his sword and says:

This physic but prolongs thy sickly days. (III.3)

Hamlet at first believes that the ghost is his father, and is quick to vow vengeance. Yet, later he worries that the ghost is in fact, the devil, trying to coax him to commit sins.

This mirrors his apparent waffling throughout the play on acting on his need to avenge his father's death. His  doubts on the identity of the ghost are just like his doubts on his ability to kill Claudius.