If you start with Cloth armor,5 HP pots and W you have to start on Ancient Golem with a leash. Follow on to Wolves, Wraith, Double Golem, and Ancient Lizard. Now you should be level 4 and near full health to gank. Look for ganks. You can always just attempt a gank you know you're going to fail to just burn their summoner's spell. If the jungle is respawning, get back to it ASAP because you want to get to lvl 6 because your gank potential is so much higher then.

If you start with 2 HP pots,Sapphire crustal and your E you should do Wraith Camp first (smite the big 1), then wolves and then wait 5-10 seconds

9 Greater Mark of Resilience
9 Greater Seal of Evasion
9 Greater Glyph of Shielding
3 Greater Quintessence of Fortitude

Starting Items 1 Cloth armor and 5 HP pots

Starting Items 2 Sapphire Crystal and 2 HP pots ( for more experienced junglers)

Ninja Tabi if facing 2 or more heavy physical dmg dealing champs like Caitlyn,Vayne,Ashe,Corki and so on. If facing heavy magical dmg dealing champs go for Mercury's Threads, But u should get
Heart of Gold and a Sapphire Crystal.

Mid Game:
Ninja Tabi or Mercury's Thread
Heart of GOld
Aegis of Legion
Catalyst the Protector
Negatron Cloak If the have heavy heavy magic damage, DO NOT finish your banshees. Instead buy a second , potentially even before finishing your catalyst.

Late Game:
Ninja Tabi or Mercury's Thread
Heart of Gold
Aegis of Legion
Banshee's Veil
Abyssal Scepter
Sunfire Cape

Keep in mind that u have to change some items depending on the enemy team.

http://www.own3d.tv/video/135533/Dan_Dinh_s_Amumu

A really helpful video for everyone.

Ghost-Smite
Ghost also has its advantage. When you Ghost at a group of enemies with your allies very close behind.

Flash-Smite
Flash is a good because it allows Amumu to flash over walls or brushes then combo with an ult and bandage toss

In team fights, you want to position your self so that you are in front and your allies are behind you. Make sure your range is about 600-900 range behind you and your melee is almost side by side with you when you are roaming. Being close means that your allies can follow up immediately when you've caught someone.

As Amumu, after the initial engage you want to tantrum and despair but delay your ultimate until you see the enemy starting using their burst on either you or an ally. You can either use your bandage to single out 1 person if they are all running.