There is a creative writing section on this website. However it's very difficult to get readers on this website because each article should be stand alone so it would be very difficult to connect your chapters to each other. This website is directed more toward articles and your creative writing pieces may not become featured making it even harder for readers to find you.
I would strongly recommend writing your book in full and getting it edited. You will find you want to change a lot of stuff in the beginning once you get to the end. It's free to publish through Amazon's createspace (or there's a very small fee. I can't quite remember). You can then sell on Amazon and potentially make money.
If you really want to share chapter by chapter I would recommend the website Wattpad. I know there are other websites specifically for writers to share books piece by piece, I can't remember any others off the top of my head.
Either way you're going to have to do your own marketing to find readers.