I honestly would not risk messing up my kindle by downloading a third part program.
Chess on Kindle
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I used that program for quite some time and had a lot of fun with it. Then I dropped my Kindle and it's screen cracked. You never know what will damage your device most, third party progs or poor handling. 
If you own an Amazon Kindle Keyboard e-Ink reader, you can play against popular engines (like Fruit, Stockfish, Crafty, etc.) on it. Somebody managed to run this program on Kindle 4 Non-touch, too. It's completely free and open-source. Here is the link: http://www.mobileread.com/forums/showthread.php?t=152815