"Sign in to Kindle Cloud Reader
To buy and read Kindle books, sign in to Kindle Cloud Reader with your Amazon account.
From a web browser, visit Kindle Cloud Reader.
Sign in with your existing Amazon account (or create a new one). Any existing titles you previously purchased will appear under the Cloud tab. You can read directly from your web browser and don't need to download an app.
Unable to Open Kindle Cloud Reader
Kindle Cloud Reader signed in with the Wrong Account
About the Latest Version
Kindle Cloud Reader is compatible with the following browsers.
Google Chrome 20 and higher on Linux, MacOS X, and Windows.
Mozilla Firefox 10 and higher on Linux, MaxOS X, and Windows.
Safari 5 and higher on MaxOS X, and Windows.
Safari on iPad with iOS 5 and higher."
You can use Linux on a device with a keyboard to read books purchased from Amazon and Calibre or something else for epub. As far as I know there isn't yet an Amazon app for reading purchased books offline