If you feel that the number of  apps on the Amazon App is not enough for your Kindle Fire, you can get access to  Android Market. The guide below will help you do this: 

1. Root your device using this guide or here.
2. Transfer this and this .apk files to your Kindle Fire via PC / Mac (remember where you put them)
3. Open Root Explorer App, and navigate to the transferred .apk files
4. Select the GoogleServiceFramework.apk and install.
5. When complete, click “Done” and return to the file location for Vending.apk
6. Long-press Vending.apk and select the “Move” option on pop-up menu
7.Navigate to /system/app folder
8. Change mount option from RO to RW (read-only to read-write)
9. Paste Vending.apk into the /system/app folder
10. Long-press Vending.apk and select “Permissions” from pop-up menu
11. Change permissions to match all other apps in this folder (rw-r-r-)
12. Reboot your Kindle Fire
13. Once reboot is complete, select Market App and log-in as usual.

If you’re looking for a way to access the Market via Amazon’s customized UI on the Kindle Fire, check out the mod posted by XDA member munday.