"处私刑杀害" can be translated into English as "kill by lynching" or "execute by mob justice".
"Lynching" refers specifically to the illegal killing of someone, usually by hanging, without a legal trial, often motivated by racial prejudice, hatred, or a desire for vigilante justice.
"Mob justice" implies a form of punishment or execution carried out by a group of people acting outside the law, often in a violent or uncontrolled manner.
Both expressions convey the idea of an unauthorized and often brutal form of punishment leading to death.