Using Kindle on your Chromebook does not require an app of any additional software.

Go to read.amazon.com and you will be greeted with this page

Now log into your account, after successful login you will be greeted with this dashboard

From here you can select a book and start reading, it will look something like this

If no books appear or it says your library is empty, go to the Kindle Store and purchase some books. There are also free books on the Kindle store for you to 

Happy reading!