As your question implies, serial killers in our society are assumed to exhibit some set of characteristics that might make them more readily identifiable than those in the general population--in other words, if we identify the characteristics most often associated with anti-social and, more to the point, murderous behavior, we may be able to find and stop serial killers early in their careers to prevent horrific and innocent deaths. Law enforcement agencies, with the help of sociologists, psychologists, medical doctors and cultural anthropologists have developed various models to help define potential serial killers, and yet, human dynamics--that it, exceptions to the rule--complicate the process in many analyses of serial killers--in part because white male serial killers exhibit different attributes from white female killers and non-caucasian serial killers are different still. For the sake of consistency, I will focus this discussion on the largest group of serial killers which, because of its numbers, has been studied more thoroughly than others, the white male serial killer.
A very common myth associated with serial killers is that, on the whole, they all suffer some form of serious mental illness. As one study has noted, however,
While there appears to be no mental illness, they definitely transgress the accepted mores of society, have personality disorders and distorted visions of the world around them. They are commonly seen as suffering psychotic and eventually sociopathic behavior. . . . (https://www.academia.edu/2578330/Profiling_Serial_Killers_A_stratified_look_at_common_characteristics_spread_across_taxonomy_race_and_gender)
Psychotic and sociopathic behavior, of course, is part of a normally adjusted person, of course, but it does not reach the level of mental illness as that is usually defined and is usually idiopathic--that is, it affects the person, not the outside world.
Setting aside human dynamics for a moment, white male serial killers have several traits that may lead them to the ultimate expression of inhuman behavior, but these traits are not universally present: in an analysis of six of the most infamous serial killers--from Sam Berkowitz (the "Son of Sam") to Theodore Robert Cowell ("Ted Bundy"), with an average age of 28)--(i)four were known to torture animals; (ii) five to have had no father figure in their early lives or to have been abandoned by father or mother; (iii) five of six either had had IQs or had education beyond high school; (and iv) four of six had some history of substance abuse). Although it is difficult to extrapolate from these facts that serial killers are always negatively affected by one or more of these attributes in their lives, it is difficult not to conclude that the desire to torture animals, which is the ultimate exercise of power over the helpless, or a history of abandonment, which leads to a different kind of helplessness, can lead to both psychosis and/or sociopathic behavior.