The good news is that all of this is possible, especially with the Kindle Fire’s source code being released, together with a one-click root method that was shortly discovered. The change of UI might not be as appealing, especially since Amazon pretty much went out of their way to change the UI to a custom one of their own that they claimed would differ from other Android tablets, but if you really want to install Android Market and source your apps from there, you can find the instructions over at BriefMobile.
Once again we have to mention that all of this is done at your own risk, so if you feel that this would change your tablet more than you like, perhaps you might want to stick to Amazon’s app store. However if the idea of being able to access both the Android Market and Amazon’s app store appeals to you, knock yourselves out!