I'm going to assume for a moment that you are an aspiring mystery or crime fiction writer and that you are okay taking tips from someone who is not. That's going to make all this a lot easier and much less weird. There are five things every serial killer character needs: motive, victim selection, mode of death, trophies and body disposal.
Motive is kind of important. Beyond how things go down, this is why your fictional killer is the way he/she is. A tip: if your motive is "because he wasn't loved", you'll need to paint in a few more colours.
Victim selection is part of what keeps the police busy. They want to know why and how these specific people were chosen. Again, it needs depth. Opportunity helps, but there needs to be something twisty to it. Something that leaves the detectives staring at a whiteboard and getting nowhere until other pieces fall into play.
The mode of death that your killer chooses should be related to the motive. It's the outward expression of what he/she is trying to do, but is somehow incomplete.
Trophies are obviously what is kept to remember the victims. To some, they might be memories of the kill, but to others they may be a way of holding on to what the victim meant. More like a keepsake than a trophy, in that case.
Finally, how the killer cleans up after him/herself. Some would hide what they've done, ashamed and unable to face it. These are more likely to have keepsakes. Others like to taunt police, flaunting their alleged intelligence. They're more likely to make a display. When all else fails, think of motive. It all comes back to that.
Mokalus of Borg
PS - Sometimes I've considered trying to write a book.
PPS - Nothing ever gets finished, though.