What are the characteristics that a champion should have to jungle?

You can actually "jungle" with anybody who is able to full-fill these requirements:

1. Clear the jungle in a decent time and with enough health left, so you can efficiently gank the lanes and don't get behind in gold & experience.
2. Be able to duel the enemy jungler in case of counter jungling. This is crucial in higher elo levels, because you will get counter-jungled 8 out of 10 times in platin or higher. (for example: Amumu is considered a decent jungler in lower to mid range elos. In high elo play he is a dead fish in the water because he can't duel till lvl 3-4 or later. Decent teams will abuse that to make you fall behind.)

And thats it. The rest is up to your preference. Keep in mind that you only truly jungle during the laning phase, after that its team-fights and free roam ... In other words: Your skillset should be be useful in this phase as well.

I disagree with some other answers, "great ganking potential" is simply optional. When Shyvana was becoming popular in the professional scene, it was a shock to many people because her gank potential is terrible. However, she had a fantastic clear speed and great mobility, making her great at taking her opponent's jungle camps and therefore denying them experience.

A jungler needs:

1. The ability to clear jungle camps (relatively fast to be useful as a jungler)
2. Survive the jungle, which many champs have a hard time with, even with runes.

Strong base armor or health stats can help your survivability, and many champs have passives or abilities that either:

• A. Give them health back per hit/ability, extending survivability.

  B. Deal area of effect damage, making it easier to clear creeps quickly.

A jungler should be able to:

• Clear the jungle fast
• Be able to survive the jungle (obvious, but important point)
• Be able to gank well

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

I created my own metrics for calculating the efficiency of junglers.

I grade every aspect from 0 to 10. Than add them up and compare junglers. The grading part can be subjective especially if the person doesn't have enough experience in the jungle.

The metrics are:

• Lvl1 fight: The ability to fight at lvl1, amumu cant do anything, while riven can deal massive damage or aatrox reviving. Add these: (AoeCC=5, aoedps=4, sustaineddps=3, somecc=3,bushcheck=3,aura=2,utility=1)
• Lvl2 gank: The ability to gank at lvl2, this wins solo ques, you need to throw people offguard with a lvl2 gank. xin zhao would get 10 in this field. Add these: (Easy to perform CC=5, Harder to perform CC=3, Displace=5,High damage=4,Sustained damage =3, Easy dash=4,Hard dash =2,Soft CC=2, Chasing potential=2)
• Lvl3 gank: The ability to gank at lvl3, some champions can not gank before 3, some can gank at 2, but their 3rd skill is useless. Riven for example, good at lvl1, better at lvl2, even better at lvl3. While a master yi gets heal at 3 which does nothing for his ganks at 3. Add these: (Easy to perform CC=4, Harder to perform CC=2, Displace=4,High damage=3,Sustained damage =2, Easy dash=3,Hard dash =1,Soft CC=1, Chasing potential=1)
• Lvl6 gank:The ability to gank at lvl6, some champions get in god mode at 6. Warwick lvl6 gank is pretty much guaranteed kill, udyr 6 isn't gonna do much difference than 5.
• Mid-game teamfighting: presence in mid-game teamfights. Bruisers like leesin, khazix shine in this part, 2-3 man teamfights.
• Late-game teamfighting: presence in late-game teamfights, aoe like malpite, amumu shine in this part.
• Jungler clear time: 10 = Fast aoe, 8 = good aoe, 6 = slow aoe/ no aoe high damage, 4 = good damage, 2 = ok single damage
• Buff clear time: Ranking from fastest to slowest
• Dragon clear: 10 = Early solo, 8 = mid solo, 6 = late solo, 4 = can tank, 2 = meh damage
• Counter jungle: Add 2 for each ability, Dueling, Fast clearing, Escape, Assasinating, Extended vision, Crowd control, Sustain
• Sustain: 10 = Always full life, 8 = Usually full life, 6 = Shield and damage reduction protection, 4 = Clears fast, 2 = Some damage reduction

Here are a few results: Kayle: 7,9,9,8,8,10,10,8,8,10,6 = 93 Amumu: 3,4,6,10,10,10,8,8,6,4,6 = 75

I am a diamond solo, challenger team jungler from s3 and ex semipro player.