In general, what characteristics should a jungling champion have?
A jungler should be able to:
Another thing to consider is how well they can invade the enemy jungle, and how well they can deal with being counter-jungled.
Some junglers specialize in one area (Ex. Alistar is not the best at clearing the jungle but can gank VERY well). Other junglers can do everything good, but not the best (Ex. warwick can clear the jungle good but not the fastest and can gank but has trouble with it until he has his ultimate).
Here is a tier list of junglers used http://rog.clgaming.net/tier-lists/46-jungle/latest
Here is a GREAT guide that can give you more information http://www.mobafire.com/league-of-legends/build/who-is-your-jungler-and-what-does-he-do-114118
For the most part a good jungler has 3 out of these 4. (With a few exceptions e.g. Maokai).
When these are factored in you can get a powerful jungler.