If you're still ever prowling around, Myrmidon, I must say I like this idea a lot. It's certainly very unique, and I'm like a kid in a candy store when there's plenty of room to capitalize on the story.
I have a bunch to say about this, but I'll start with general archetypes as to why someone might want to die or kill. You have several of them, but I'll list them anyways and maybe something I say will speak out. Also note that there are a lot of grounds for overlap.
1. Imprisonment / Torture. If someone knows they are imprisoned / eternally tortured in such a way that they will never escape, death might be something welcome. Alternatively, they could want their jailers / torturers dead.
3. Perceived weakness. As part of a weakness in themselves they resent (or what you represent to them) they might want to kill you.
4. Death of a loved one. This would obviously not be in the initial group of people you'd encounter, but could definitely be a piece of the puzzle later in the game. Imagine that someone the protagonist killed was quietly loved by someone from behind the scenes, or a retrospectively repenting family member. They might want revenge on you or to be killed.
5. Lack of a future. This ties in to disability but is painted in a different light.
6. Disability. If someone is incapable of living out their lives normally, they might ask themselves why they should live.
7. Outcast / Banishment. Tied to loneliness, but painted in a different sort of light. Society has shunned the character, and they will be alone forever.
8. Intense emotional trauma. What happens when someone is raped, for instance, and cannot cope with being the mother to such a person? Trying to avoid as much political thunder as I can here, but it's a thought to consider.
9. Honor. Imagine someone from before the time the legend who was mortal, and specifically has been looking for a way to die in battle. Perhaps in their religion in order to reach an equivalent to heaven they must die honorably.
10. Justice or Revenge. If someone had a score to settle, felt wronged, or actually was legitimately wronged, they might want that someone dead.
11. 'Baby Hitler' scenario of murder. What if you know someone will grow into a madman and will bring suffering into the lives of many?
12. Evil. Tying into justice and revenge, if someone was legitimately perpetrating evil, it might be in everyone's interest to see them dead.
Your character (and his entourage), will be "blessed" with the curse of mortality. They can die, but they will be the only ones in the world capable of killing.
This is a potentially very rich plotline staring you in the face. How did this happen? Are there hidden gods that decided humanity should not be as it is, creating the protagonists as their instrument? Did some archvillian believe this? Is someone doing this a scare tactic to obtain power, withholding what he knows can destroy you so he can do so at the right time? There's a lot of ways you could go with this.
Another thing to consider is how the protagonist feels about each influence and kill. Honestly, I feel like this story would benefit greatly from direct dialogue pertaining to this. If you don't want to do that, you could still reflect on how the character feels in subtle ways in the environment around them, but this would be trickier.
EDIT: Something else I should say... be careful not to go too overboard on the morbid in the game. It is clearly supposed to be dark, but be careful to put in a good comedic relief now and again. Basically, don't desensitize your audience; it'll ruin a lot of the effect of a game like this. When you have only dark moments, there's no contrast. When you have a lot of moments of good, it puts the darker moments into perspective. A good example of this is one you should know: Disgaea.