What crimes does Dogberry accuse Borachio and Conrade of? What does Don John do after Hero is accused?

I think that you are talking about what happens in Act IV, Scene 2 of the play.

In that scene, Dogberry is basically accusing Borachio and Conrade of being liars.  He calls them "false knaves."  In addition to this, Dogberry says that they have taken money in exchange for lying.  He says that he heard Borachio saying that he had gotten a thousand ducats from Don John for lying about Hero.

We also find out that after Don John hears about Hero being accused, he runs away.  We are told this by the sexton.