"处私刑杀害" can be translated into English as "lynch and kill" or "carry out a lynching and murder".
"Lynch" specifically refers to the illegal and violent act of punishing or killing someone without a legal trial, often by a mob.
"Kill" is a straightforward term meaning to cause the death of someone.
When combined, "lynch and kill" or "carry out a lynching and murder" accurately convey the meaning of "处私刑杀害" in English.