I tried to install composer via brew per:
In usr/local/bin (which was not on Mavricks and I had to make personally) I did.
brew tap jose
The "Getting Started" page is the introduction to the documentation. Most documentation will start off with installation instructions, just like Composer's do.
The page that contains information on the composer.json file is located here - under "Basic Usage", the second page.
I'd recommend reading over the documentation in full, so that you gain a better understanding of how to use Composer. I'd also recommend removing what you have and following the installation instructions provided in the documentation.