I'm mentally ill and I really, really dislike that trope, too, but it's not really possible to be a person that goes out of their way to kill people multiple times without something going on that could benefit from going to therapy. That doesn't mean he has to have bipolar, borderline, psycopathy, schizophrenia, or any diagnosable thing in the DSM5, but he will
have something going on that is causing him to process his emotions, desires, or view of societal norms differently than a "regular" person. It might be a concussion, it might be a traumatic event in his life, it might be being raised in a culture/setting where life isn't respected, but there is
something that caused him to go down that path.
There's a few different things that can make a serial killer memorable:
- The number of victims. The more you kill over the shorter period of time, the scarier that is
- The types of victims. If it's homeless people/sex workers/drug addicts, then people (including the police) are going to not care all that much, but if they're young pretty white girls then people are going to lose their minds. At least one serial killer had his victim returned to him by the police because "oh well that's just what gay men do, I guess!"
- The way they kill or dump bodies. Uniquely mutilated bodies, being found in the middle of the desert, "calling cards" left on the bodies. This goes into....
- Their "brand." We remember the Zodiac Killer because of the puzzles he sent to newspapers and the mystery he built around himself, even though he only killed 5 people. Sometimes the "branding" is created by police/newspapers and builds this mythology of who they are
- The person themselves. Serial killer clown! Really hot guy! A regular looking nurse at the hospital! If they look really scary/creepy or they look incredibly normal or even attractive, that really draws people, too, based on what people assume evil people to be (i.e. ugly)
It's usually a mix of these things. The scariest thing one can come up with is in your own imagination, so don't go into a bunch of gory details if you don't feel comfortable doing it, but let your readers fill in the gaps themselves. Their brains will come up with something that is incredibly scary to them, so take advantage of that.