Never one to be passive-aggressive, Clifford comes right out with it when something's got his goat.
In the very first scene, he says, "Let us assail the family of York" (1.1.66)—right to their faces. We appreciate his gusto and frankness, but is it too much? Clifford is so bent on getting back at York for killing his dad that he kills York's defenseless son Rutland. A head for a head? Killing a young kid and his tutor to get back at the kid's dad is pretty cruel. We don't want to cross this guy anytime soon.
So, let's get this straight. York killed Clifford, Sr. (back in Henry VI, Part 2). Clifford killed York. York's son Richard wanted to kill Clifford and certainly would have if Clifford hadn't died off stage first. That's a revenge cycle, all right, and this little drama of family vendettas is like a mini-version of the big wars going on in the play. There may be more people involved, and there may be a lot more bloodshed, but it's really just a family squabble in the end.