My 5 year old son threatens to run away or say that we don't love him or say very disturbing, self-deprecating things when he doesn't get what he wants. How do we deal with this?

Has there been any major change lately in his life? Like arrival of a sibling, change of place or school? Your kid needs a lot of your attention and time now. Something is bothering him else such a young lad shouldn't be saying all these things. Keep on talking to him lovingly, don't scold him always. Try to prod what's going on in his mind. Love heals everything.