As mentioned earlier, revenge is one of the engines propelling the novel forward. Whether insults are real or perceived, characters take immediate steps to redress the issue. No one in The Three Musketeers likes getting hurt; when they are injured or insulted, they make sure they get even.
Milady exacts her revenge by using others because she lacks the physical strength to exact it by force.
Revenge has nothing to do with Athos’s determination to try and sentence Milady. He is motivated purely by justice.