What should a son-in-law do when his father-in-law dies?

Actually, this needs to be based on the local customs of your father-in-law's side, because some places do not allow mourning for sons-in-law, so the best way is to ask your wife to ask them what the local customs are like, but As a son-in-law, you can just follow your wife's direction. In fact, you can follow them in whatever they do. Your father-in-law is just like your own father.