It would be useful to know which model of Kindle you own ... However, I believe that regardless of which model it is, you should be able to do what's called "sideloading": temporarily downloading the eBook to an intermediary computer (desktop PC, laptop or netbook) and then transferring the book to your Kindle via USB cable. For this, you don't install any special app; you can use PCLOS's file manager (Dolphin, PCManFM, Thunar, etc.). In other words, you don't need a native Linux program on the intermediary computer to read the eBook because you're simply using the computer's capabilities to transfer the file to your reader.
Although I don't have a Kindle (I own a Nook Simple Touch), I use this method all the time to transfer Project Gutenberg eBooks from my PC to the Nook.
If you need to transfer your purchased (DRM-ed) Kindle books from your PC to your Kindle, you might try the procedure outlined here:http://www.kindleboards.com/index.php?topic=58105.0
(See esp. the final post in that thread, by user pidgeon92)
If that doesn't work for you, I read that the Kindle Boards forum is a great source for Kindle-related help:http://www.kindleboards.com/index.php
If you haven't checked out Project Gutenberg yet, I highly recommend it. They have an excellent selection of totally free eBooks in the public domain, in both Kindle and Nook format. You'll find enough to keep you happily reading for a long time ...http://www.gutenberg.org/
HTH and Good luck.