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?

Such a young child would not say such things unless something is really bothering him. You need to talk to hum and find out what's going on in his mind. External influences would tend to make him use such threats. Give him extra attention and affection so that his insecurities die down. Change will surely come.