[Blossom]

If you have an Android device with Market Access you can install ES File Explorer and it will backup any apps from the Market into a apk file then just upload to Dropbox and Install on your Fire.

To backup an app while in ES File Explorer.
Hit Menu/Manager/App Manager
Select the Multi Select button at the top
(It's the one with the + on it.)

Select apps to backup by hitting each one in the File list.
Hit Menu/Backup to backup the apps.

Or Long press on one app and choose backup.

You will find the apk files in the folder backup\apps.

You can then upload them in ES File Explorer using the Dropbox app.

Long press on the apk file a window will popup,
Scroll down while in the Operations window and choose Share\Dropbox\Send to This Folder.

Once that's done, on your Kindle Fire where you have sideloaded the Dropbox app.

Open Dropbox and find where you sent the apk file
Then just press on it and It will open the installer automatically.

Note: You can download the Dropbox app directly from their site but Silk blocks it. So you will need to do this via USB or email it your email. (Not the Kindle's email!) However I recommended side loading Opera Mini you can download apk files directly from a website with it.