How to get two AI civilizations to go to war against each other?
Solution 1:
When in the diplomatic/trade interface with other leaders, one of the options you can expand is "Other Players" - and in this submenu, there are options for asking for a declaration of war on another civilization.
Bear in mind that this is one of the more costly things you can ask for of an AI Civ, and they'll often refuse ("There is no way to make this deal work") or ask for exorbitant fees.