Now that the Kindle Fire is rooted, you can actually install the android market on your Amazon’s tablet and use it the way you do on any android device. Downloading of apps is successful, as confirmed by mashi and MOPhiPsi.

If you are new to Kindle Fire development, it’s imperative that you have rooted your Kindle Fire (for which you need to set up ADB first, btw!). Installing ADB, getting ADB to work and then rooting the Kindle Fire is a bit tedious right now, but it’s been nicely explained here. So, get it rooted first and then use the guide below to install android market on your Kindle Fire.

Root is required so that you have full access over the OS, getting ability to write/transfer required files to the core of the system. Once rooting has been done, check out the well-written step-by-step  guide quoted below — it was written by XDA member, Jolleyboy.

Not only the Andorid Market, this guide will get you many Google apps, and some other cool apps too.

Jellyboy’s guide:

This download includes:

Google Services Framework
Android Market
Books
Voice
Maps
Google+
Reader
Gmail
Street View
Youtube
Talk

As well as some goodies that everyone needs:

Netflix
Pandora
Dolphin Browser
Go Launcher

Without further ado: The Instructions.
1. Download http://www.multiupload.com/DGMBZZNOXM onto your computer. This includes all of the Google Apps, as well as a few other free apps from the android market that you might like to have on your device. (GO launcher is required to access Android Market at this time because the launcher included with your fire doesn’t show it with your other apps)
2. Connect your Kindle Fire to your computer via a micro-usb cord.
3. Create a unzip the contents of the downloaded link onto your Kindle Fire (You will need WinRAR, sorry I was too lazy to zip it. If you don’t have WinRAR yet, get it. It’s amazing. )
4. From the Amazon App store, download “ES File Explorer”
5. Open ES File Explorer and navigate to where you unzipped those .apk files.
6. Select “GoogleServicesFramework.apk” and install it.
7. Restart your Kindle Fire
8. Re-Open ES File Explorer and install each of the apps.
SPECIAL INSTRUCTIONS for ANDROID MARKET

Before you install the Android Market you need to copy the app into /system/app and set the correct permissions after it is in /system/app.
You will need to chmod your android market apk to set your permissions. Your permissions should be rw-r–r–, the same as the other system apps. http://androidforums.com/behold-2-al…rmissions.html has a good post explaining what these mean and how to set them up using chmod. Rather than going into detail here I will refer you to this link.
Props to Bandage for identifying this
If this whole process is too complicated for you, instead you should sideload Titanium Backup which can convert the market apk to a system app. This method should work as well.
Props to Bongostl for figuring this one out.

9. Restart your Kindle Fire
10. Open Books or Gmail and log into your google account
11. If you have any errors past this point Restart your Kindle Fire and everything should work out fine.
If at first you have difficulty getting the Android Market not to force close (after restarting) go to settings -> applications -> third party and find Market on the list. On the market’s setting screen clear data and try again.
I hope this helps you in your quest to get Google Apps on your Kindle Fire. If you need any more help post a comment and I will help you as quickly as I can.
UPDATE: If you still have problems with your android market use this file instead http://forum.xda-developers.com/atta…2&d=1321476329
Update: http://munday.ws/kindlefire/MarketOpener.apk is an app made by Munday that puts a market Icon in the kindle ui and opens market after you have it installed. This saves you from using another launcher and is very helpful. Props Munday!

Special thanks to Jolleyboy!