The serial number for a Kindle is located in the settings on the device. Alternatively, you can find the serial number on Amazon.com by clicking on the drop-down "Your Account" menu and selecting "Manage Your Content and Devices."
The first thing to do is make sure that you registered your Kindle is registered to your Amazon account, if you're going to attempt to view the serial number online. Otherwise, the model of Kindle may determine what steps you actually need to take to view the serial number. The serial number is listed in the information about the device within the settings, but the path to accessing that information varies. Amazon.com has a number of helpful articles and tips about navigating your Kindle device, sorted by Kindle type.