Full AP Amumu with amazing damage thatcan jungle very easily. Change smite to ignite/exhaust if not jungling. Only do this build if you have a good advantage either in the beginning, or champ composition or enemy is just bad.
Similar starting items, but this build is a lot more tanky, but has 40% CDR and still decent damage.
If you are new to jungling, this guide has jungle routes covered:
Jungling refers to the act of having a character not stay in the lanes, but instead travel around the jungle, taking out neutral minion camps. When done correctly, jungling can level a champion as quickly as a solo-laner. This lets your team have, essentially, three carries - the solo champion in middle, the solo champion in top, and the jungler - all of whom level extremely quickly.
As a jungler, your job is to take out these camps while not dying (unless playing Smite+ Revive Evelynn, but that's a conversation for another time). Most importantly, after securing your buffs and being reasonably high leveled, your job is to gank the mid and side lanes.
Higher-level games will always have a jungler, as the advantages are too good to pass up. As such, when your ELO rises and you notice more and more enemy junglers in normal games, you need to be able to deal with them. For now, however, this guide will take you through how to jungle with just Amumu, not how to counter others.
When starting off, you will need to assign roles - who is the mid carry, who is the top soloer, and who will be dual laning at bottom. Try to assign your support characters, like Taric, Janna, Soraka, or Sona to the bottom lane with melee characters. Try to assign characters with lots of staying power, like Mordekaiser, Akali, or Nasus to the top lane.
Start off at the patch of grass between your blue buff and mid lane. This allows you to scout for an early gank at your own golem and prevents first blood. If wanted, you can have your team stand guard with you as well.
Go to the blue buff golem and take him out with your Despair. Level up Tantrum and move on to the wolves. Take them out with your W and E. In a similar manner, kill the wraiths, making sure to autoattack the big blue wraith first.
Then go to the Lizard buff and kill it with your W and E, Smiting at around 470 health. Finally, destroy the small golems. This lets you level again, bringing you to level 4. Put a point in your W if you want to continue jungling. If not, upgrade your Bandage Toss and gank.
Leashing is simply the act of having a teammate stand on the other side of the wall by blue buff and doing a bit of damage to it, whether it's from a spell or an autoattack. This will cause the blue golem to start trying to attack your ally. When it loses aggro, it will switch targets to you - but during all that time, you've been attacking it and crying by it.
Counter-jungling is a term that refers to when a champion enters the enemy jungle and attempts to cripple the enemy jungler by stealing his creep camps (denying him experience and gold) or just killing him outright.
The most common way is the first - simply steal creeps. Smite can kill the blue Wraith at level 1, so a lot of junglers will go and Smite the blue wraith, then run to their blue buff and jungle normally. Other junglers might kill a Wraith and a Golem, while some can steal red buff out-right. Counter-jungling requires a good knowledge of different junglers, their routes, and their timings.
As a jungler, your job is to gank as many champions as possible early game, preferably their mid carry. As a tank, you should be expecting to get assists, not kills. As a tank, you also have the right to initiate a tower dive, but only at full health or close to full health. Try to start ganks as soon as possible - ideally, you'll be in a position to gank right after grabbing the small golems and porting back to base, but more realistically, you'll start around level 5 or so.
Remember to protect your carries late-game and that your ultimate should be prioritized as a defensive positioning skill over an aggressive push - if it comes down to you dying or your mid carry dying, sacrifice yourself. Of course, not dying would be optimal.
You are the main character on your team responsible for warding and grabbing Oracle's Elixir. With your gold per 10 items, it's basically just you and your support champion who can afford to buy these items without slowing down your item build. Constantly buy wards, constantly ward for your teammates, and mid-late game, try to have an Oracle's active on you most of the time to eliminate your enemy team's wards.