I appreciate you asking this question because I think the issue with "mistreatment as backstory to antagonist behavior" in writing is … it's tropey, which isn't bad when tropes are tools to tell a story with, but it can be tiring when it comes off as, "He was treated badly so now he's crazy."
I'm in a small online community for people with diagnosed Cluster B personality disorders—not because I have those diagnoses or symptoms myself, but because the administrators do allow non-personality-disordered people to join and hopefully the neurotypical (or neurotypical-adjacent? neurotypical-passing?) people gain insight and understanding from their honesty.
And I used to be part of a different community that considered mention of any association between destructive behaviors and personality disorders to be discriminatory stigma that must be silenced in an act of alleged social activism, because of…so many paragraphs of jargon…that did not show in the conduct of the authors at all. These were abusive people who explained their harmful behavior as a disorder while at the same time denying that it was harmful in the first place, or denying that they had to do anything about it (recognize, make amends, deescalate, prevent similar harm in the future…at least get therapy) … because they tended to rather use their disorders as a shield from criticism or as a shield from the voices of their (usually also-minoritized) victims. I think it's giving them not enough recognition of their support needs (or too much credit) to say they "used" their disorder in that way: the behavior pattern suggesting this was there, but the deliberation might have been inhibited.
While in the first group I've mentioned, I have read descriptions of personal experiences (however much you can trust those online, but I'll mention it anyway because I think it could be true) that went like: "The way I am doesn't come from trauma, but I do want to be a different way than I am because people keep expecting that of me, but…" just many examples of Not Doing That, which sounded to me as though there's a very formidable barrier to bridging the present reality and an improved future. This individual didn't seem to know how to try to improve, not even when specific actions have been explained with words by the non-disordered people in this disordered person's life.
I interpret this as kind of the nature of these disorders, that what's obvious to everybody else often isn't on the same wavelength as the designated patient, so communication doesn't get "translated" from, like, English to English—let alone internalized or even really comprehended. The mechanism of the disorder, the way it operates, gets in the way.
But more importantly to your character, the disorder doesn't always come from trauma…but sometimes it does, and in those cases it's easier to understand even if the occasional reaction is, "I had it rough too but I don't act like that, plenty of people in the world have had terrible childhoods and don't act like that…" which is fair as a boundary against the suggestion that allowing somebody to hurt the speaker would somehow heal the one doing the harm (it doesn't work that way, so anything to shut that idea all the way down is good: nobody wins the trauma olympics.)
However, there's entirely different internal circumstances and processes in the "having it rough" that makes the difference between some people who don't turn into murderers…and, some that do. Which is different from people who don't develop personality disorders, and those who do.
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I keep thinking there should be different words for "neurotypical"; there are trait mental illnesses such as depression that increases mood irritability or fatigue, and those can take therapy or time/seasons or medical interventions—and there'll be some good days that depressed people can think "I'm cured" or better yet not think at all but simply live their life without hypersomnia or out-of-proportion anxiety getting in the way.
There are also character disorder personality disorders, which are different from trait mental illness in ways that make the former difficult to treat but aren't defined by the inability to be treated—and so the real difference has been wonderfully presented in ways I'm not sure of. If I recall, personality disorders affect perception, but then so can depression at times.
Straus & Kreisman's book I Hate You—Don't Leave Me put it better, although that's more focused on advocating for the humanity and rights of sufferers with Borderline Personality Disorder…while at the same time being low-key queerphobic, for those who might be interested in reading this text but would prefer to be forewarned of those parts.
So this is all to say… Do you really need your character to be "crazy"? Because there are so many different kinds of "crazy" that can lead to the same result—and from my layperson knowledge and (if I may say so myself) unusual experiences, each of those options would take as long to explain as this post: pushback after CPTSD ("battered wife syndrome" except it's more like "battered son"), or a highly symbolic set of hallucinations, or a subclinical lack of emotional affect, or literally textbook twisted moralism as in maybe he read something somewhere and took it to heart in an entirely illegal way that he didn't only do something excusable but he saved the world from the person he murdered or something like that.
So if you write an otherwise neurotypical person in extraordinary circumstances that led them to do what they did, as in to say, "You—neurotypical person, yes, you—You could be exactly like this character, if you were unlucky enough in life!" Then that's a different approach of intertwining fictional character traits with plot events that develop the character—such as murder.
So I think there's an entirely different schema to approach with understanding what a personality disordered fictional character would feel like to be personality disordered—and then the story (or backstory) involves murder.
Really, I think either way is fine in the brainstorming stage before you decide to pursue the way of writing it that you're going to pursue. And there's plenty of in-betweens or another axis of approach.
