This is my personal guide to jungle tank Amumu. This guide outlines the steps, build, and strategy for succeeding with Amumu. Your focus is providing team support, initiating fights, constant enemy harassment, and serious cc. Kills are just the bonus.
As with any guide, it is simply a guide. Skill and knowledge combined with multiple team factors can make or break any game. Play smart. Pick your battles and prosper.

Beginning steps:
Standard rotation on jungle. Start with blue buff. Leash will be needed. Team should watch area for potential attack. Next kill wolves, specters, and then red buff. Always check bush near red buff for counter jungle. Recall back and buy philosophers stone.

Next head to area with potential gank. Your grab (Q) is key. A proper timed grab can be the death sentence for any champion. Let characters with flash or cleanse burn their abilities. Don’t tower dive unless enemy has extremely low health. Champions with low magic resist will be easier targets to solo.

If you cannot successfully achieve a gank then continue to farm jungle. Wasted time means lost experience and gold. Boots are next priority. How soon and what type will be dependent on your kills and what you’re up against. Ninja tabi for team with heavier auto attack / ad. Mercury treads for team with heavier ap / any cc. The first build is for heavier AD team with the second being for heavier AP team.

At this point continue to focus on potential ganks, cover lanes when needed, and power farm to level 6. Once you have your ultimate punish anyone foolish enough to be deep in your lanes. Don’t be afraid to use your ultimate at earlier levels to mop up single and double kills. Also properly timed Ultimate's can save your teammates from certain death. You want the other team to fear you. Heart of gold is next priority item in the list.

You should be picking up blue buff as soon as it respawns. This will keep your mana up for fights and harassing. Recall back only when at low health or have downtime. Keep farming levels and gold. Sunfire cape is the next item on the list.

Once turrets start to fall, team fights will become more common and become the focus. Initiate fights once your team is ready. You want to be in every team fight possible. Save your ultimate until needed at this point. Use downtime to push turrets, farm creep / jungle / buffs, and work on your build. Use the build guide for item purchase priority. Everything after this is self-explanatory. You goal : win