Some savvy hackers figured out a way to get the Android Market working on the Kindle Fire. Although this makes the tablet far more of a direct Android experience, keep in mind that not all apps on the Android Market will work on the new Kindle Fire. Still, that will start to change soon enough, and then you can get to using your Kindle Fire how you choose to. You have to be rooted to get this to work, but after you accomplish that, here's a set of instructions for the process:
Prerequisites
  • Root your Kindle Fire
  • Download the following apks (You are on your own finding them, Id rather not link them here): GoogleServicesFramework.apk & Vending.apk (Latest Version 3.3.11)
  • Root Explorer App (or similar that lets you change ro/rw permissions)
  • Side Loading Apps activated on Kindle Fire
  • On the Kindle Fire Settings screen, go to "Device" and turn On "Allow Installation of Application From Unknown Sources"

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

Let us know in the forums how things went if you give this a go, and as always be careful and proceed at your own risk.

Source: Kindle-FireForum via AndroidPolice