In Act 3, Scene 1, Polonius, Ophelia and Claudius hatch a scheme in an attempt to disrupt Hamlet’s pursuit of Ophelia, and this passage contains alliteration.
Immediately before Polonius and Claudius hide, Polonius advises his daughter to read a prayer book in order to seem more natural as Hamlet approaches her. He muses that people are often blamed for faking religious devotion in order to cover up their sinfulness. Claudius, who is doing that very thing, is affected by Polonius’s offhand comment and reveals—as an aside to the audience—the extent of his emotion, saying: "O, 'tis too true / How smart a lash that speech doth give my conscience." Claudius’s aside is a rare opportunity for the audience to see how he's processing his guilt. The importance of his speech is emphasized by the alliteration in this passage, as the /t/ sound is repeated in quick succession in the like "O, 'tis too true."
In this way, Claudius uses the inherent musicality of his language to draw the audience into his confession of guilt. He affirms the truth of Polonius’s statement with emphatic language. In fact, Claudius uses alliteration quite a lot; as a character, he is drawn to performance because he lives in a lie. The audience comes to expect him to put on airs and long speeches as he attempts to live up to the role of king. This instance is unique in that Claudius is communicating only with the audience. His admission begins with an alliterative edge, which pulls the audience into his guilt and forces them to feel the sting of his actions as he reflects on what he has done.
In Act 1, Scene 2, Hamlet discusses his grief with Gertrude and Claudius. After Claudius makes a long speech about the need to move past mourning the previous King, he and his new wife interrogate Hamlet, whose sadness is evident and therefore a threat. Claudius encourages Hamlet to move on, promises to love him as a father loves his son, and requests that Hamlet not leave Elsinore. Claudius says:
And we beseech you, bend you to remain
Here in the cheer and comfort of our eye,
Our chiefest courtier, cousin, and our son.
Claudius uses alliteration in this phrase, repeating /b/, /ch/, and /c/ sounds throughout these three lines. To understand the value and purpose of this use of alliteration, it is important to remember that Hamlet was written to be performed more than read. The repetition of these sounds enhances the texture of the language itself, drawing the audience in by using the same consonant sounds over and over. This use of alliteration is meant to enrich Claudius’s speech, exacerbating his persuasiveness by giving it extra rhythm. It is clear that Claudius is pouring effort into easing Hamlet’s tension and distress, and the sound of his language makes the audience pay special attention to these phrases. However, because Claudius’s requests are at odds with Hamlet’s emotional reality, the eventual effect of the phrase is one of emptiness. It seems that Claudius’s communication is mostly performance and not very heartfelt, considering how elaborately he tries to convince Hamlet to stay and behave himself.