This is a build I have been using for ages and you can become very dominating late game, if you play well and land amumu's skillshots succesful. This build will work very well in teamfights, you will deal constant damage and have some usefull passive's like Frozen Heart, Rylai's Crystal Scepter and Abyssal Scepter too help out your teammates.

I start with a Cloth Armor and 5 health potions, just like many junglers start their build.
I prefer taking early swift boots so I can keep up with my enemies in early ganks.
As you should have your first kills and assists, take a small trip through the jungle until you can buy an Abyssal Scepter. It will make your jungling alot easier. As enemies start dealing a bit more damage aswell I start making amumu more tanky with items like Giant's Belt, Chain Vest and Negatron Cloak. Upgrade these items too Abyssal Scepter and Sunfire Cape and buy and buy a Rylai's Crystal Scepter, which slows your enemies while you deal magic damage and making it easier too chase your enemies. Frozen Heart and Athene's Holy Grail will bring your cooldowns with 35% and give you extra ability power, armor, magic resist, mana and mana regen. Including Frozen Heart's unique passive, slowing down your enemies attack speed by 20%.