Kritzekratze is probably understood by most German speakers as an ad hoc binomial onomatopoeic construction based on the Inflektiv of the onomatopoeic verb kratzen. [All weird words in this sentence should become clear later on.] In addition, it can be understood as an allusion to Wilhelm Busch, a precursor of modern comic book authors who died in 1908 and a very important figure for German popular culture. I believe most Germans still know the following lines from Max und Moritz by heart:
Max und Moritz, gar nicht träge,
Sägen heimlich mit der Säge,
Ritzeratze! voller Tücke,
In die Brücke eine Lücke.
In Busch's case, ritzeratze was the sound made by the saw which his (anti-)heroes used to make a trap for their victim. Whereas ritzeratze is purely onomatopoeic in the sense that the verbs ritzen and ratzen don't make much sense in Busch's context, kratzen has a fitting meaning, as it translates to scratch.
The verb form kratze is more interesting than it would appear. The Inflektiv is a recent invention normally not taught to students of German, or in fact to anyone. Apparently, it was first developed by translators of English comics, who were presented with the hard problem of coming up with translations for verbs used as interjections, as in "BRAKE! SQUEEAAAK!!!!! ... SIGH! SOB!". Later, it seems to have been popularised by Erika Fuchs, a famous German translator of Disney comics. (She certainly is to blame that I grew up with the Inflektiv.) By now the Inflektiv is simply the verb form used in German comic books in this context. People also use it in internet chats, and occasionally you will even hear someone say something like seufz, ächz, stöhn! when they feel they are having a typical Donald Duck moment.
Technically, the Inflektiv of kratzen should be kratz rather than kratze. But it is normal for German verbs to have optional -e endings in some cases (e.g. ich kratz[e], ich hab[e]), where the tendency is to lose the -e more and more over time. And it is normal to restore an -e that one would normally drop if it makes sense for high register, rhyme, metric, or, as here, onomatopeia and allusion to a famous literary passage.
The alternation of the stressed vowel (often between the standard one and a contrasting one) is a form of binomial that appears to be a bit more popular in German than in English. In English I can think of ding dong, ping pong, tic tac [toe], zigzag, singsong, snip snap. They all have German equivalents, but German has some more that are done differently in English. E.g. English for Hickhack is argy-bargy; Tingeltangel is honky-tonk.