First of all I will give you a list of junglers, some aren't the best ones, but they can jungle:
Lee Sin, Udyr, Dr. Mundo, Nocturne, Jarvan IV, Maokai, Nautilus, Skarner, Trundle, Alistar,Riven, Shyvana, Olaf, Amumu, Malphite, Cho`Gath, Shaco,Warwick, Shen, Hecarim, Jax, Gangplank, Master Yi,Volibear, Rammus, Sejuani, Fiddlesticks,Darius,Xin Zhao,Wukong,Nunu,Tryndamere,Singed,Heimerdinger,Evelynn,Twitch,Akali,Karthus,Sion ,Fiora.
Some of this champions can make you say, what the fuck ? But they can jungle and can be good. I'll give you some examples.
If he has the blue buff he can do the jungle really well , and having a ganker with slow and extreme damage output can be very good, of course you wont get that farmed, but you can get fed, since with your ultimate you will just kill people in no time. You will start your jungle with his Q.
To jungle with him , and with most champions you will need runes of armor at yellows, you can go with flat magic pen at reds, flat ability power at the quintessences and ability power per level in blues.
You start at wolves with damage then you need a pull at blue, after that go wraiths and get back to wolves, after that do red and you can try to gank that it is overpushed.
The downsides : Well you won't get as much farm as you could in mid, Karthus isn't that powerful in late game. It is a bit hard to farm with him in jungle, and he has some hard time ganking.
Singed is by nature a tanky champion so he can be useful in a team comp. He definitely needs to start at blue, he needs mana so you can use your Q to do the jungle . After Q you will be getting E, so you can farm even more easily. You should go with runes of armor at yellows, flat or per level magic resist at blues , magic pen at reds and flat ability power at quintessences. His main purpose is his ganking ability , just go in with your poison use fling on the target, or if he tries to flash you use the slow and then fling it , if he doesn't flash before fling , you fling and then you slow him . And it is an almost certain kill.
The downsides: He is a slow,slow jungler... He can be countered hard in jungle, just steal his blue and he is fucked!
These are just two examples of not so much used junglers.
So I will be talking about the jungler role in general, I won't give you guides about each champion, and how you will play them in jungle. Maybe if I do guides on each champion , the ones that are able to jungle I will make a part for it .
So has jungler what will you do ?
You won't be in a lane farming , instead of that you will be in jungle , killing the neutral camps and helping your teammates through ganking. And don't ever forget! Take smite!!!! Always take smite, it isn't just so you can do your buffs more easily , it will help you securing your buffs, counter jungling, and the most important, securing Baron and Dragon.
Most of the times your items of choice will be : Cloth armor + 5 pots . If you are with a strong jungler you can take vampiric specter, boots of speed + 3 pots , or even with a Doran's Blade.
Ask mid to give you a leash on the buff that you start , if in blue ask top to help you in blue by giving some damage and protection, if in red, ask bott to come and help you the same way as it was in blue.
After that you will proceed to killing the other camps, since the new jungle , it is useless to do the Double golems, so don't waste time there, at level 3 you should be with red and blue buff. And now you should start ganking.
To gank you must first look for lanes that your adversary is pushing, after you acquire your target lane you should start going there and always ask that lane if they had warded it. And warn the lane that you are going to gank! Most of the times that people don't warn the gank is a supreme fail! You should wait for the laner to start , or if you have a stun you should start, depends on the situation, and always remember! Making your adversary to use flash is a good gank!!! Don't be too greedy or you will die! There are alot of tactics to gank, I will give you videos of some ganks so you can see, it is hard to explain through text.
What you should do :
You should allways try to give your buffs to your teammates that need them more, for example blue for AP mid. You should control your jungle and theirs, how you do this ? Through warding! Always but always ward baron and the buffs , if someone tries to steal your blue or red if you have it warded you will just get a free kill with the help of your mates!
You should start jungling when you want, but take it easy , jungling without runes is really harder, but it is not impossible, you should just play it safer .
When possible ward the river to protect your teammates ! Having that ward will not only help them not being ganked but will help you in ganking, now you can see if they have warded the place or if there is an enemy there.
Try to buy oracles elixir or vision wards, it will improve your ganks since they will no longer have wards to protect them.
Also, when a teammates asks for a gank try to go for it.
And don´t forget! There are two ways of jungling , with ganks or without ganks!
Farming the jungle:
The only difference in this one is that you wont be doing many ganks but instead you wont leave the jungle so you can farm, not to used because it gets you underleveled and it is hard to gain gold from this , it is easier to get it from kills or assists provided by ganks.
Another important topic is counter jungling! What is counter jungling?
Well counter jungling is when you steal xp and gold from the enemy jungler! For example if you saw that the enemy jungler, just failed a gank at top , or is top you could go and do the main wraith! How will you do that? You enter their jungle, if possible ward it , it will be useful , and just do some damage on the main minion and then smite it and get back to ganking or to your jungle.
Why just the main minion ?
Because you need to be fast, so you just damage and smite, also he is the one that gives more money, and because if you don't kill all the minions in some camp it will not respawn , so for example you steal their blue , you can leave one minion and if they don't go there and kill it their blue won't respawn.