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?

First of all spend some time with him daily to go to root of the problem. Some directions are as follows: - Kids at this tender age tend to imitate the things they learn from surrounding. TVs or friends might be a source. - He might actually feel neglected. How are you spending your time? Do you have another kid who you are favouring unintentionally? - He might want what other kids in his circle possess and that might be a source of comparison. You might need to think about some positive reinforcements to encourage good behaviour by rewarding him reasonably.