In general, Hamlet's responses are convey through language in several ways. First and most generally, he externalizes and verbalizes his response. That is to say, things that might stay inside of you or me, as feelings we don't have words for, Hamlet spells out, loudly and in detail. Second, and closely related, he provides metaphors and images to help with this. Third, Hamlet "over communicates"—he gives responses that explain the past history of the situation.
(Oh, if you're looking for Hamlet's response to something or one specific, you'll want to supply a little more context. What event or scene in particular intrigues you?)
In Hamlet's response, we can found out a great deal about word-play at its best. Shakespeare had used them beautifully and elegantly, using and combining numerous of multivalent terms in various ranges, from outright gross puns to highly-nuanced words of diverse metaphorical terms and images, associating and personifying objects (for example feelings, emotions he can't literally describe to the audience), that is displayed wonderfully by using literary devices like imagery and allusion.