Answer:
That depends on the type of vampire. Vampire types range from the very ethereal to animalistic to humanoid. An energetic or demonic vampire would probably be the least likely to be able to kill it's equals but also the one least likely to be self aware enough to "want" to. It may not even have instincts to. The type most "human" is also the one most likely to be emotionally self aware with human tendencies toward violence. There are kind of three different basic types of vampire. Energetic, animalistic, humanoid.
Demons and energetic vampires, like wraiths, may not be able to affect one another much less kill each other, and why would they. No competition to survive, not as acutely aware of itself/others.
Animalistc would perhaps have more reason to kill each other, competition, territory, instinct of survival of the fittest.
The more human the vampire is, the more acutely it feels emotions. The other side of that coin is that the more acutely a vampire feels emotions the darker side of human it becomes.