"Before my own God, I might not this beleive, without the sensible and true avouch, of mine own eyes." - Horatio

This particular moment shows a disbelief for the existence of ghosts - they would not have believed it to be true had they not witnessed the ghost with their own eyes, because the existence of ghosts goes against their religious beliefs. However, seeing the ghost and confirming it is real, leads them to think - still in the religious ideal - that it's a ghost from purgatory, who's either stuck there to do sins, or unfulfilled business.

"I am thy father’s spirit,
Doomed for a certain term to walk the night
And for the day confined to fast in fires,
Till the foul crimes done in my days of nature
Are burnt and purged away."

This conversation implies that Hamlet's father is stuck in purgatory - a religious ideal. It is a catholic belief that there exists heaven for the good, hell for the sinful, and purgatory for those with finished business. However, it leads the readers to wonder... why is Hamlet's father's ghost asking him to avenge him, and commit the murder of Claudius? Ghosts in purgatory don't usually ask for fulfillments like that...
Purgatory, as defined by the Catholic Church, is: "purification, so as to achieve the holiness necessary to enter the joy of heaven,"
This would imply that one is being given a last chance to purify themselves and gain entry to heaven. However, the ghost's request is contradictory of his religion. "If you forgive men when they sin against you, your Heavenly Father will also forgive you. But if you do not forgive men their sins, your Father will not forgive your sins." (Matthew, 6:14) This quote alone proves that the ghost's request alone will prohibit him from entering heaven, as he has not forgiven Claudius of his sins - so he will not be forgiven this sin. "Thou shalt not kill." (#6 of the Ten Commandments) Despite Hamlet's father not directly killing anyone, he is ordering his son to kill another - therefore breaking the Ten Commandments and barring entry to heaven for himself.
The fact that this play is so religiously based makes this whole situation very suspicious and contradictory.

"To be, or not to be, that is the question."
Through much of the play, Hamlet is depressed and toys with the idea of suicide. He considers killing himself, however, being the catholic that he is, he is turned off of the idea due to the fact that it is a sin to commit suicide. Ultimately, this is the reason that he does not end his life. And how different things would have turned out had he not had his religion to keep him from committing suicide; he would have been dead at the beginning of the play, and then there would be no plot.
"O, that this too too solid flesh would melt
Thaw and resolve itself into a dew!
Or that the Everlasting had not fix'd
His canon 'gainst self-slaughter!"
Also, in being religious and knowing he cannot end his life, he contemplates the nobler deeds - what would be better of him to do.
"Whether tis nobler in the mind to suffer
The slings and arrows of outrageous fortune
Or to take arms against a sea of troubles
And by opposing end them?"
Although, this in itself is another contradictory aspect of the play - it may be the nobler deed to live in his pain, but if he's going to stay alive and then murder Claudius, he's going to be committing sin anyway.

This aspect of the play is up to interpretation - according to the narration done by Gertrude, it would conclude that Ophelia's death was purely accidental. However, there is reason to beleive Ophelia may have committed suicide. This suspicious causes the clergy to only do the bare minimum for Ophelia's burial, as the social and religious stigma attached to suicide would suggest that she deserves no more.

In the beginning, Hamlet treads lightly with the ghost - as it is against his religion to take advice from ghosts, he does not hasten to fulfill the ghost's demands. "The spirit that I have seen may be a devil, and the devil hath power a pleasing shape." He first decides to prove his uncle's guilt, before making any moves against him. This shows strength in his character and religion, as he could have let himself be overcome with anger from the knowledge of Claudius killing his father, and acted irrationally, by killing him. Instead, he waited to prove his guilt.
Once proved to be true, he still stalled and hesitated on the murdering of Claudius. His reasoning is open to interpretation - but it can be believed that he hesitated in fear of what would come after killing Claudius. He'd be committing premeditated murder, very much a sin.

Religion being considered a major factor in the play can be justified in this scene alone. Hamlet, intending to kill Claudius, finds him at his most vulnerable state - in prayer.
Had Hamlet not been religious, the whole play would be completely different - because he would have killed Claudius then and there. However, he was of the belief that Claudius was in prayer, asking God for forgiveness for his sins, and therefore would have gone to heaven if Hamlet killed him then. So, he resists.
Unfortunately, considering how things went, had Hamlet killed Claudius then, it would have changed the fate of the seven others who lost their lives (Polonius, Ophelia, Hamlet, Laertes, Guildenstern, Rosencrantz, Gertrude).

"O, my offense is rank it smells to heaven;
It hath the primal eldest curse upon't
A brother's murder."

This is a direct reference to the biblical story of Cain and Abel - Cain murdering his brother Abel being an analogy to Claudius murdering Hamlet Sr.

"Fie on ’t, ah fie! 'Tis an unweeded garden
That grows to seed. Things rank and gross in nature
Possess it merely. That it should come to this.
But two months dead—nay, not so much, not two.
So excellent a king, that was to this
Hyperion to a satyr. So loving to my mother That he might not beteem the winds of heaven
Visit her face too roughly.—Heaven and earth,
Must I remember? Why, she would hang on him
As if increase of appetite had grown
By what it fed on, and yet, within a month—
Let me not think on ’t. Frailty, thy name is woman!—
A little month, or ere those shoes were old
With which she followed my poor father’s body,
Like Niobe, all tears. Why she, even she—
O God, a beast that wants discourse of reason
Would have mourned longer!—married with my uncle,
My father’s brother, but no more like my father
Than I to Hercules. Within a month,
Ere yet the salt of most unrighteous tears
Had left the flushing in her gallèd eyes,
She married. O most wicked speed, to post
With such dexterity to incestuous sheets!
It is not nor it cannot come to good,
But break, my heart, for I must hold my tongue."

Hamlet frowns upon and is perturbed by the remarriage of his mother to Claudius, not only due to the sense of betrayal for her replacing his father, but a sense of moral and religious wrongness. It is not prohibited that widow's remarry, however for her to remarry so fast would be frowned upon. In addition, she is marrying her brother, which is incest. Incest, which IS prohibited by the Catholic religion.

"Not a whit. We defy augury. There’s a special providence in the fall of a sparrow. If it be now, ’tis not to come. If it be not to come, it will be now. If it be not now, yet it will come—the readiness is all. Since no man of aught he leaves knows, what is ’t to leave betimes? Let be."

This is the moment where Hamlet reveals to Horatio he's going to let the chips fall where the may, and accept that God has chosen his fate for him. If he is to live, he will. If he's to die, he will die. This is a direct reference to Matthew 10:29; "Are not two sparrows sold for a copper coin? And not one of them falls to the ground apart from your Father’s will."