Well, I presume the more esoteric death cults and so forth are up to the creative force behind the cult in question. But within the background of the big four, lessee...
Callidus- Would need to be fairly personable for their modus operandi. It's all very well to disguise yourself, but if you can't display a wide range of emotions you aren't going to get very far. That said, I imagine a callidus assassin may well have some identity/personality issues- you spend so long behind so many masks, you're going to become pretty existential about the 'real you'. I could see a certain amount of see-sawing in behaviour, alternating between acting and responding in ways that are most likely to garner the trust and affection of their colleagues, then snapping back into being really distant and cold, to try and compensate for behaving as though they were 'working'. I also imagine trust would play a big part in any relationship or friendship they formed- living behind a disguise being so important to the assassins survival, it would be incredibly meaningful for them to abandon that safety net- to show someone their 'real' face.
Eversor- Well, these guys are indeed hampered by being drugged up rage machines. I guess again it depends on their fluff- as in, are the drugs permanent, or mission dependant. Because if they have periods of lucidity, then that can be really interesting. You could have an individual who shakes and scratches from the withdrawal, or someone who is totally dissociated from the personality that emerges when he works. The idea of a calm, polite, skull faced killing machine appeals somehow. Perhaps he oil-paints... Do they remember what happens when they kill? Are they directing the slaughter, or do they ride along as a helpless observer of the carnage? Or is their concious mind totally oblivious, but the experiences of their subconscious self always boils beneath the surface. I believe it is an Conclave members signature: “I would be king of infinite space, but that I have bad dreams...”
Vindicare- These guys actually seem to be almost bland, given that their methods involve killing from a very long way away. It would be easy to go down the cold sniper route, but that seems generic. As immensly skilled individuals, I imagine that they would have superiority complexes from here to Cadia. I would imagine it is tremendously hard to be someone who could shoot a penny from two miles away, and not develop a certain condescension for the little people. I could almost see them as giant, human cats- arrogant, aloof, having servants rather than friends, and if they screw up they would immediately kill all witnesses and pretend it never happened. I doubt they would lick themselves in the same way, but apart from that. Alternatively there could be a patronising element, trying to impart the ways of their obvious talents upon poor, benighted lessers.
Culexus- Imagine being someone that no one could love. Pariahs are soulless, and inspire revulsion in their peers, but there is no reason that they couldn't have the potential to be psychologically rounded human beings. It could be truly tragic- wanting to be friendly and warm and caring, but never receiving validation for the overtures you make. Bitterness is certainly one option, but another is over-reaching- A Culexus with no concept of personal space and boundaries, who always tries to start each mission with a group hug, and gives you inappropriately expensive gifts out of the blue, and they have absolutely no chance at all of being endearing. Almost makes you feel sorry for the abomination.
Is this the sort of thing you had in mind? I mean obviously there is no limit to how characters could be conceived, but these struck me as some interesting rough profiles that moved beyond the 'cold focused killer' model.