If you don't want to spend a dime, then your best bet would be to customize your CentOS build to include a VNC server, and to run that server during the installation process and every time the computer is run. This wouldn't be the easiest method, but it would mean that once the installation had a network connection established, you could connect to the computer with your laptop using a VNC viewer.
If you have any friends, you can borrow a mouse and keyboard to perform the initial CentOS installation, and then use the same basic instructions I gave you a link to, to install and setup VNC, and then return the keyboard and mouse to your friend. Then, you need only connect via a VNC viewer from your laptop.
If you want to head to your nearest Salvation Army or Goodwill store, you could purchase a keyboard and a mouse for less than $10 USD, and then use them to install and set up CentOS. You can still install VNC, and put the keyboard and mouse away for future use.
Personally, I recommend getting the keyboard and mouse, and having them should anything go wrong in the future. But, if you insist on not getting them, then the first option, customizing your installation media, would have to be the way to go.
EDIT you could also just perform an Unattended Install of CentOS. You would have to find instructions that fit your tastes, and follow them. In most cases, they would involve a little tailoring, but otherwise you would create the install media, connect the install media, start the computer, and stand back. Like...
or just do a search for "unattended CentOS installation" and/or Kickstart.