think of CC as providing the collection feature that Kindle Fire does not have. on KF you see your 100s of books all jumbled into the books folder and you cannot impose any structure on that.
I was horrified, when I got my KF HDX to find that there is no support for collections, even though the e-ink Kindles have it. lots of negative comment on amazon discussion re this but it seems that Amazon have no plans to ever add collections to KF- they'd rather you bought a 2nd Kindle device for your book collections!
CC uses wireless connection so there's no need to ever connect KF directly to calibre via USB. you do need to get your amazon books into calibre though , before you can use CC , & you will get help with that in the main calibre thread.
I read of an alternative called shelfari that gives you virtual shelves for all of your books at amazon, but I think it's only for .com users and I am .co.uk.
But if all your books are bought/borrowed via amazon that may be better for your needs, as you don't then need either calibre or CC
http://www.shelfari.com/
( its says "by amazon.com" so maybe they own it ?