What you are trying to do is called sideload ing. Before doing this, you will need to go to Fire's:
"Settings" => "+ More" => "Device"
and turn on
"Allow Installation of Applications"

Then you can simply open the APK from the email client and install it in almost any file manager.
ES File Manager is commonly used to download applications to Fire.
Here you can find out more about using Dropbox to press the APK on your Kindle, and which browser to use instead of the built-in silky gizmo.
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