When I'm picking a victim, I am almost always looking for good cities to take. I plan to keep at least half of the cities I take, almost all of them if my economy can handle it.
Factors I consider, sort of in order of precedence:
1) Distance - they have to border me. The only exception is if I'm moving onto another continent, or my war is to cripple an AI that is a tech or space race threat.
2) Who can I safely attack? I'm not going to launch a war against Mansa Musa if an annoyed Alex borders me. Aggressive, back-stabbing Civs who border me usually receive highest priority. It's best to attack stronger Civs first, but not if they are so strong you're not sure about the outcome. You can always build yourself up taking weaker neighbors before taking them on. You also have to see what their diplomatic relations are with other Civs. Attacking a Civ that is friendly with other Civs may result in having to fight more than Civ at a time. Obviously, you have to be aware of defensive pacts.
3) Quality of cities and territory: What do they have in the way of resources, cities that have founded a religion, shrines, wonders, etc.? Happiness resources are often a priority, especially in the early game. Religions are important, since I usually take them rather than found them.
4) Are they about to get a UU? If you have the opportunity to axe-rush Caesar before he gets iron, that's hard to pass up.
5) Is there any possibility of demanding techs for peace?
6) Who am I going to get diplomatic penalties with? Is there any possibility of getting resource trades cancelled as a result? Will I end up losing an important trading partner?
