The Cockroaches love causing chaos wherever they go. They seem to enjoy watching people suffer, whether be simply humiliating them or outright torturing them. The Cockroaches seem to lack any kind of moral standard; They willfully commit crimes on a near-daily basis with no remorse whatsoever. This is most prominently shown in "Soldier for a Day" where they accidentally destroy the entire world along with everyone but show no regret in doing so.
Their relationship with Oggy is very odd. They have no sympathy for him and they laugh every single time they witness Oggy being put through all degrees of pain. The Cockroaches practically live off on Oggy's suffering, with nearly every episode in the series involving them trying to ruin Oggy's life in some new degrading way. Despite this however, they secretly need him in their lives. In Copy Cat or Oggy Robot, when Oggy seemingly gives up on his life, the Cockroaches begin to grow bored of their own lives and attempt to spice it up again with a robotic clone of Oggy. And just like Oggy, they begin to lose their minds without each other after time passes. In "Moving Out", the Cockroaches are at first ecstatic upon finding out that Oggy has moved out, passing it off as a joke. However, when they find out that he is truly gone, both Marky and Dee Dee begin to go insane without him, crushing themselves repeatedly with Oggy's fly swatter just to relive their memories together. When Joey finds out the location of Oggy's new home, they do everything in their power to reunite with him (which horribly fails).
The Cockroaches also show a deep sense of family. As abrasive as they are, they truly care about each other (Caviar on the House! being the most notable example) and in Roachy Redneck or A Roachneck Cousin, they're extremely happy to see their cousin visiting them.