Submitting to the Amazon App Store is very straight forward and their documentation is very easy to follow.
First, log into your developer account in the Amazon App Store dashboard or create an account if you don't have one. Once you log in, you'll be brought to this page where you'll create a new Android App.
Next, you will be brought to the screen below where you will have to enter general information about your app. Don't worry if you plan to make a change in the future or if you are unsure of what to put right now, you will still be able to edit this after you click 'Save'.
After you click 'Save', you will be brought to this dashboard where you will fill up the Availability & Pricing, Description, Images & Multimedia, Content Rating, and Binary File tabs before your app is submitted for review by Amazon.
On the Binary File(s) tab you will have to click on 'No' when Amazon asks to sign your app. Note: If this step is not available in your Amazon Portal, you will need to contact Amazon customer support to ask them to enable 'Self Sign Status' for you. You can contact them using their 'Contact Us' form. After they enable it for you, this option should now be made available.
This step is required for every Skillz game you upload to the Amazon App Store. This step will also add 5 additional steps to your Amazon submission process. Here are those steps:
Step 1: Upload your Binary into Amazon Developer Portal
Step 2: Download the binary that you just uploaded using the 'Download Binary' button
Step 3: You will upload the binary you downloaded from Step 2 into the Skillz Developer Portal and click Save. Next, you will have to click 'Publish' on the 'Store Ready' column of the binary that you just uploaded. Note: Clicking 'Publish' will make your app available for download in the Skillz Store.
Step 4: Download your game's apk in the Skillz Store by going to http://games.skillz.com/game/YourGameID using your computer. i.e http://games.skillz.com/game/1410
Step 5: Upload this Skillz Signed binary you downloaded in Step 4 to Amazon Portal.
Step 6: Republish the original binary that you uploaded to Skillz, so that the non-Amazon binary is available on https://games.skillz.com.
After completing these steps and filling up the necessary details in the binary file(s) tab, you can now move on the last tab of your Amazon Portal which is 'Release Notes.' After inputting your release notes, you are now ready to submit your app for review!
Here is the link to the official documentation of the Amazon App Store.
Note: Besides the Amazon App Store, you can also host your Android games in the Skillz App Store which you can learn more about here.