Angst-ridden, irreverent, detached and matter-of-fact
A Clockwork Orange is almost a foreign-language work: it's not written in British, American, or standard English. Instead it features nadsat, a made up language incorporating elements of Cockney and Russian spoken by the "modern youth" in the book.
Figuring out what Alex means with each term is a feat in itself, and it takes a few chapters for even the most astute reader to get a firm grip on the language. Now, once you think you've crossed the language barrier, the tone will be easy to gauge. Alex is a matter-of-fact kind of narrator, although he does embellish some of the goings-on for dramatic effect (this shouldn't be surprising because nadsat employs a decent amount of onomatopoeia, or the use of words that sound like what they mean).
Much of the tone is irreverent and immature-sounding. We also detect considerable angst, not surprising given the subject matter being described. Interestingly enough, however, the tone Alex uses when describing violence might be described as almost detached. He matter-of-factly recounts exactly what punches he throws and just how much blood oozes out from his victims' orifices.