I didn't invent either of the two characters, but as I wrote dialog for them I had a few ways I liked to think of him.
Personally- and I know this was not the way he was officially conceptualized and was not actively pursued in the way the game was written- I always thought of Psychopath as actually being sociopathic (and by that I mean actually having an Axis II personality disorder, most likely Narcissistic personality disorder with features of Antisocial Personality Disorder) and wanting to inflict pain as a primary form of actually being able to feel emotion (otherwise, he has a tendency to feel either numb or dysphoric, agitated, unsettled, and contemptuous towards most of his peers), with the war simply providing an outlet for those impulses that otherwise he may have expressed in potentially criminal ways. I saw him as being someone with dark impulses who was right on the boarder of being a "villain," but having him on your side. I toyed around with the concept of him being aware that he was only a few steps away from being a serial killer, and that opened up the possibility of him either not wanting to become one so desperately trying to contain those impulses towards humans but letting them loose on Kilrathi, or simply having him see the war as an opportunity to fully express his impulses and feel fully alive. But, we never really went that far. Those were just some of the concepts I found myself bouncing around in my head with regards to the character, but I thought he had the potentially to be a very complex, disturbing, and interesting wingman.
Avatarr, on the other hand... once again he wasn't my creation, I just worked with him to some degree. In Saga's final form, he is actually toned down a lot... a whole lot. To be honest, since I didn't create the character, I never really got past conceptualizing him as a mere womanizer. I always wondered if the character was really that shallow, but I liked to think it was a facade he put on, a piece of bravado as a way to keep himself from being troubled by the horrors of war. I felt that there were one or two missions were we got to see deeper into who he really is, as opposed to what he shows the world.