I am going to assume you are playing from Zone 1 to Zone 10 and then doing a ritual to reset back to Zone 1 as there is no need to go beyond simply reaching Zone 10 with the current achievement list.
It's not possible to get 25 champion kills in one "run" of those 9 zones, but thankfully this achievement is cumulative. Just play from Zone 1 to 9 and then use a ritual on Zone 10 to reset at the "boss". You should be able to get enough kills in a few run-throughs.
Note that you will be able to do enough damage to reach Zone 10 without any upgrades, however they do make it easier and quicker.