My Brother has been falsely accused how should he best clear his name?

Yes, he definitely has a case for slander. It has cost him professionally, personally and financially. He needs to pursue this. Make sure you print out all the facebook posts first. As for costs, I'm not exactly sure. Contact a lawyer for a consultation. There are many variables, with lawyers costs, court costs etc. I hope he gets his name cleared, and his ex is exposed as the manipulating, soul destroyer she is.