Power the Kindle off. Use a fingernail to pull the back away from the Kindle just a little in the area of the lower half of the Kindle. I cannot say all models of the Kindle will open the same way, but I expect they might.
Step 1: Use a coin
Step 2: Removing the back
The process of attaching the back again is the reverse of removing it. Arch the back as shown in the photo from step 2. Secure the upper catches by sliding the back upward (Ignore the red arrow.). This sounds easy, but it is easy for one catch to remain unsecured. See the third photo.
Press the lower half of the back into place until the catches snap and are secure.