There may be several ways to install Android Market apps, but here's one way I did it.
1) In my Android Phone (Running Android 2.2 Froyo) I backed up all the apps to my SD card
2) Transferred the apps to Dropbox online cloud
3) Downloaded to my Kindle Fire and installed apk files from there...
Tools I used:
ES File Explorer (Available free from Amazon Appstore), ASTRO File manager, Dropbox
Good news, many Android Market apps work in Kindle Fire!!
Although I have Google Music app working flawlessly on my phone, I can't get the same app version online streaming on my Kindle Fire. All the apps that require Google account doesn't seems to fully work. I can use the Google services like Google docs, calendar etc from the Amazon Silk browser.
LauncherPro does not work. But Go Launcher EX is working perfect replacing the default Kindle Launcher. (The version on Amazon App store is old. Get the latest version from Android Market) Now its like a cool tablet and will generate a huge market for $200 price range tablets. I'm pretty impressed so far. More of my experience coming soon...
A better way to mass transfer apk files is through connecting to your PC/laptop. Kindle will connect through a Micro B USB cable to your PC. You might already have one if you use a Samsung/Blackberry phone. My Logitech mouse charger is Micro B USB cable and I could connect Kindle Fire to laptop via that cable. So copy the apk files to the Kindle Fire storage and start installing one by one.