Kindle Fire Tablets (3rd generation + higher) and Fire Phones
Press the Volume Down button + Power button at the same time and hold them for a moment. Your screen will flash briefly and your screenshot will be saved in your Photo Gallery.
Kindle Fire (1st + 2nd generation)
You can however simply use another device to take a photo of your screen. Just make sure that the picture isn't blurry or out of focus, and that any details are easily recognizable.