My battery is dead and when I plug it in to the wall outlet or my computer it will not charge the yellow light will not go on. What can I do?

It sounds like you simply need a new battery. Here is a site that sells replacement batteries for the Amazon Kindle 1:
Amazon Kindle 1 BatteryThe prices are some of the best around and all batteries carry a 1 year replacement warranty. I hope this helps!

Try different chargers, your yellow charging indicator should come on even if the battery is broken. It could be a broken charging port/ logic board issue as well. If you still think it may be the battery and If you've explored all options with software settings and If you're outof warranty, you can try to open the kindle and replace the battery. There is a Kindle Repair Guide that shows you how to take it apat. There are services that can do it for you. Kindle Repair.

The battery is most probably undercharged or dead, before you buy a new one, make sure that it is the battery by testing with another one in a store.