In addition to the Ghost's testimony, there is what amounts to a complete confession by Claudius in Act 3, Scene 3, when he is alone and is trying to pray. His prayers begin with the following words:
O, my offence is rank, it smells to heaven;
It hath the primal eldest curse upon't,
A brother's murder!
This long soliloquy contains many verbal proofs of Claudius' guilt, including the following explicit lines:
But, O, what form of prayer
Can serve my turn? 'Forgive me my foul murder?'
That cannot be; since I am still possess'd
Of those effects for which I did the murder—
My crown, mine own ambition, and my queen.
Hamlet does not hear any of this--but the audience hears every word. Perhaps Shakespeare intended to make his audience quite sure that there was no doubt about Claudius' guilt. Hamlet does not enter the scene until Claudius has said the following words:
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does not enter the scene until Claudius has said the following words:
Bow, stubborn knees; and heart with strings of steel,
Be soft as sinews of the new-born babe!
All may be well.
Hamlet thinks of killing the kneeling man but decides against it. Shakespeare may not have wanted Hamlet to overhear this confession because he could conceivably have found it harder to restrain himself from killing Claudius, even while he was praying. Or Hamlet might have understood from what Claudius was saying aloud that he was not really praying but only trying to pray, in which case Hamlet would not have had the same rationale for postponing his uncle's death. Hamlet only knows that Claudius is praying; he does not hear anything the guilt-ridden king is actually saying.
After Hamlet has put up his sword and exited, the scene ends with Claudius giving up trying to pray and saying these final words:
My words fly up, my thoughts remain below.
Words without thoughts never to heaven go.
The short answer is yes, Hamlet's uncle really did murder Hamlet's father. The evidence for this is in Act 1, Scene 5.
In this scene, Hamlet's father appears as a ghost to his son. When he speaks to Hamlet, the king pitifully admits that he's doomed, for a time, to walk the earth at night and to suffer the fires of purgatory in the day until his "foul crimes" are "burnt and purged away." He then begs Hamlet to avenge his death and proceeds to reveal that he didn't really die from a snake bite, as claimed. In fact, he says that "the whole ear of Denmark / Is by a forgèd process of my death / Rankly abused." Basically, the whole country's been deceived into believing a lie, and the king thinks that the lie is a horrible trick to play on an unsuspecting populace. The words that proclaim the guilt of Hamlet's uncle are below:
Ghost: ...But know, thou noble youth,
The serpent that did sting thy father’s life
Now wears his crown.
Hamlet: O my prophetic soul! My uncle?
Ghost: Ay, that incestuous, that adulterate beast,
With witchcraft of his wit, with traitorous gifts—
O wicked wit and gifts, that have the power
So to seduce!—won to his shameful lust
The will of my most seeming-virtuous queen.