Gadgets Xray has posted a how to video which explains the steps required to install Google Play on an unlocked Fire Phone. According to early user reports on XDA Forums, his trick won't work with a unit locked to a network, but the narrator in the video does mention that it works with an unroooted phone.
As you can see in the video, the process is fairly straightforward and works pretty much like the back Google Play hacks I have posted in the past for budget Android tablets.
If you would like to try this yourself, you'll need to download a set of files and install them. And apparently since Silk renames the APK install files to ZIP files, the first step will be to install a file manager which will enable you to rename the files back to APK files.
Just install the four Google files mentioned in the video in the order shown in the video and you should be able to install Play Store apps on the Fire Phone.
I plan to go try it myself (my Fire Phone is charging), and I am deeply curious about how well it will work.
Similar hacks are available for Amazon's Fire tablets, but they don't work very well. Amazon's tablets are missing key parts of Android that Google's apps expect to find. It's kinda like a watch missing gears, with the resulting app on the Fire tablet being equally functional.
Let me know if you get to it before me. I'd like to hear what you think.