There's a lot of misunderstanding as to what a serial killer is here.
A serial killer is typically somebody that murders very specific people or people that the killer feels fits the profile of people they want rid of, a 'tormentor', which is why the murder often involves personal ritual, and their motivations for doing so are always born of being 'victimised' in some context or another from their own perspective.
Not someone that just goes on a rampage. That's a spree killer. Rampages can be serial, and conversely a serial killer's exact target profile can be so hard to determine that the victims are hard to tie together and may initially appear random, but indiscriminate killing is spree killing.
That said with the aforementioned comments about a lack of narrative - a serial killer game would have to have quite a backstory.
Hitmen (Hotline Miami), hardened criminals shooting anything in their way (Kane & Lynch) and self defence (Uncharted) obviously don't apply.
I feel like Frank Black.