To become the King like a true Farmer Amumu should. You need one simple thing. MAP AWARENESS!!
You need to place a few deep wards at his blue and red so you know if he is, then you will konw if he is about to gank top or bot. Since you know what he is, you should start taking jungle camps nearby, so you can always counter gank if nedded. If you know where the enemy jungler is, you know what to do! Simple

One thing you should never do is trying to COUNTERJUNGLE. You should never try this on Amumu you will get squashed like a little crockroach. So just take that badboy to the team fights and see who will be laughing then.

When you want to team fight with Amumu which is his strenght. You need to make sure you can land the bandage toss. It's your best cc ability and if your afraid of missing it in a ranked game, then flash and ult and then bandage toss the carries. But of course you will waste your flash!

You also need to make sure when you ult you position yourself right. If you are in a "losing" fight they will move towards you then stay back so you will get as many E's out while you will slow and do damage with you W since u have Rylai's.

If you are in a winning fight the might go for a flash or try to run away. So right as you ult, you start running towards the place they will probally chase so you can keep slowing them all the way for some easy kills on your carries!

When you pick Amumu there is alot you need to be aware of.

1. Lee sin and Pantheon are your worst enemies at the start!
Your early game is more rubbish than a jungle Ashe would be. You do slow damage, you aren't tanky yet and you will have to burn most of your mana all the time. So be carefull especially in level 3 when you go for your red!

2. Pick Amumu against Fizz = NO GO!
You should never pick Amumu if you face teams with big poke since you wont be able to inniciate. This includes people like Nidalee and Fizz for being great counter picks to you.

3. Farmsimulator Amumu is on the way!
You need to go harder at farming than the midlane. You need to do EVERYTHING for gold. Line up ganks, take farm when they leave lane, clear jungle all the time when you can.

4. Fail your ult, fail your mission in the team.
Amumu's ult is so important that if you miss it, you will have to get ready for the blame tornado heading your way with 300 mp/h. Cause you have such a big impact with the ult, people will be very angry if they dont get that advantage!

When you play as Amumu you need to farm a lot as i told you! But here is the most important things to get as a Amumu jungler. And only take the things if possible of course.

Baron > Dragon > Towers > Jungle buffs > Farm > Ganks

I did it this way cause of global gold is bether than individuel gold. Also ganks are not that good for Amumu which is why it's last since you need to use your ult in most cases which you dont want to. You want to save your ultimate for the Dragon fights or ganking / counter ganking botlane!