Conveniently enough, I tested mildly positive for SPD and have been in psych studies on SPD's causes and effects. These days I'm quite practiced at controlling some of the more troublesome effects, or at least keeping people from noticing too often (it helps that my case is fairly minor).
First up, I'd like to reiterate the comments that have been made regarding the use of SPD as a label. I strongly recommend reading some reputable material on SPD to familiarise yourself with what it's actually like; it may be that your character is just a bit of a loner and/or has leftover emotional damage and/or PTSD after the abuse you mentioned. SPD is (as far as I know) rarely a conditioned disorder; that is, it's more often a thing that you either have, or don't - you don't "get" SPD because of something that happened to you. It's really worth getting this right, partly because of how much it could enrich your story to properly explore SPD and how it affects people, and partly because you don't want to fall into the trap of using "oh look he has SPD" as a shorthand for "this character is weird and misanthropic", which often happens with mental health issues in fiction.
Anyways, all that aside, on to your actual question.
Hatred is definitely within the range you could expect from a schizoid personality, but it might not manifest in the way you'd expect. SPD doesn't necessarily mean a person doesn't feel emotion - it's more that they do feel it, but don't like to (because they can't control it, because it's not rational, it doesn't make sense, it makes them do things, etc... the reasoning varies, assuming they do actually know the reasons consciously). So, they'll try to avoid situations that cause emotion. In my experience this generally leans towards positive emotions - that is, the person will tolerate something that triggers positive emotion a little further than something that triggers negative emotions. Either way, when things become too much the person will try to remove themself from the situation. This may mean physically leaving, or they might just become silent, passive, "shut down", and generally stop participating. This isn't always perceived as a conscious decision. For me, all of a sudden the people around me would feel like strangers. I could feel alone while sitting in the centre of a room filled with all my friends. In extreme cases I'd feel like I was on autopilot, watching it happen, and the autopilot just wanted to sit and be left alone.
So, hatred is definitely possible. It'd take a fair bit, because hate requires holding on to a grudge while the usual response for a schizoid personality is to detach, ignore, and distance themself from the person, but it does happen. As for that hate pushing them to kill... I don't know. Like I said, my results were mild, but a lot of SPD is about control (for me anyway) - control over yourself, how you're feeling, what you're doing. People and events can make you feel emotions; that's a loss of control. Emotions make you do things; that's a further loss of control. The response is therefore to control the situation, so that you can control what it makes you feel, so that you can keep control of everything else. If you're stuck in a situation where you can't disengage, then things can get... interesting.
I don't think I'll go into it here (PM me if you want the story), but based on my experience I don't think premeditated murder is a particularly "schizoid" way of dealing with it. At that point you're crossing a line (or several lines) over into what is probably sociopathy. However, if your character feels trapped with a person that they hate or is making them angry, then it's possible they could kill, if they thought it necessary to go that far. It's not so much that they'd lose control; more that they'd NEED to get out of the situation, and they'd see only one way to do it - it'd be a totally logical decision (from their point of view). This would take a LOT of pushing though.
Hope that helps!
