First off, good luck! Very interesting topic.
I suggest several routes:
*Research serial killers and how they behave. They all are different, but feature similar traits such as: narcissism, childhood trauma, and brain damage/illness.
*Get into the mind of a serial killer. What is your character's motives? Revenge? Sadistic joy? Etc.
*Who are the victims, if any? Is this person killing a specific type of target (many serial murderers kill targets with similar appearances. Ted Bundy killed women who looked like his ex-girlfriend)
*How much empathy will your killer have? Most serial killers show little to no empathy or remors, but some do.
*How will your character kill victims, if he/she chooses to do so? Women and men often murder in different ways for different motives. Killers often leave behind a signature mark or keep body parts as reminders.