According to a BBC News article Criminologists identified four distinct characteristics of people who kill their own family members.
The four types
Self-righteous: Killer seeks to locate blame for his crimes upon the mother who he holds responsible for the breakdown of the family. For these men, their breadwinner status is central to their idea of the ideal family. (case study: Brian Philcox)
Anomic: The family has become firmly linked to the economy in the mind of the killer. The father sees his family as the result of his economic success, allowing him to display his achievements. However, if the father becomes an economic failure, he sees the family as no longer serving this function. (case study: Chris Foster)
Disappointed: This killer believes his family has let him down or has acted in ways to undermine or destroy his vision of ideal family life. An example may be disappointment that children are not following the traditional religious or cultural customs of the father. (case study: Mohammed Riaz)
Paranoid: Those who perceive an external threat to the family. This is often social services or the legal system, which the father fears will side against him and take away the children. Here, the murder is motivated by a twisted desire to protect the family. (case study: Graham Anderson)
Take a look at the article