Unlike some Illuminati, your special ability works directly in helping you achive your goal of destruction. Focusing a Cthulhu deck It is far too easy to build a Cthulhu deck that can destroy anything but is incapable of controlling anything without an automatic takeover. It's also easy to build a deck that isn't particularly designed for Cthulhu, such as the aforementioned Violent Criminal Overlords type decks that would work as well for Bavaria. What a true Cthulhu deck needs is theme . Concentrate on destroying certain kinds of groups rather than indiscriminant killing.
Many popular decks are vulernable to one or more of the cards listed above.
The good part is, the groups that help you destroy you favorite targets tend to work well together. Gaia's Revenge Collect all the Green and Liberal groups in the above list. The emphasis seems to be on destroying Corporate, Science and Computer groups, with options on Media and Weird groups. The temptation is to increase the scope of destruction to Conservative and Straight groups by adding Fiendish Fluoridators and Society for Creative Anarchism. The danger is concentrating too much on destruction and not enough on controlling groups from your hand, or protecting your power structure from attack.