I have not seen or heard of anyone who has successfully connected a video card or USB video to the Galileo board.
There are ways of getting a Linux console. You can connect via Serial post (that Audio jack is not audio it is a Serial port connector), you can connect using SSH over the ethernet, and I think you can connect via telnet over the ethernet. You can also telnet and SSH over WiFi, but that is a little more difficult.
Even if you do connect a video card or a USB video stick, you may need to connect to the Linux console first in order to be able to setup the video outputs. I am not sure about that, I am not really a Linux expert, and even if I was, the Galileo does not run a full blown Linux image. Though some people have built one for it.