Please note, this guide is not definitive and your experience may differ slightly but this article gives you a good idea of where to start.

This process takes about one week and you have to go to the office twice, once to fill out the forms and again to pick up your extended visa.

To extend your visa you need the following items:

• A copy of your returning flight ticket or confirmation that you’ve purchased it
• A statement written in Japanese by either your kyoto/kocho-sensei explaining that you need more time to pack up your stuff and say proper good-byes (tetsutzuki in Japanese). Note, you cannot write that you want to sightsee as this is not a valid reason.
• A bank book or statement showing you have enough money to support yourself while you are here.
• A document that shows when your contract finishes (your school will have this).
• In addition is is a good idea to have your passport and gaijin card on hand.

Better information from the official Japanese Immigration Office can be found here.