How can I run a new light circuit from a GFCI outlet?
To wire up a pull string light switch you basically just need to bring "hot all the time" power to it. The same sort of unswitched power you bring most plugs in your house. The pull cord on the light will act as the "light switch". So you just need a black (hot all the time) wire and a neutral (right back to the panel) wire.

Looking at your GFCI you will notice you have it right there. Your GFCI will be labelled LINE and LOAD on the back of the plug on the plastic. Whatever wire nut the black and white wires go to on the LINE side of the GFCI is where you want to attach your black and white wires going up to your light.

So in review. Take a 14/2 wire, run it from your GFCI box up to your octagon box for your pull string light. On your lamp base put the black wire on a copper screw and the white wire on the silver looking screw. In the GFCI box, look at where your white and black wires coming off the LINE side of your GFCI are going (to what wire nut). Attach your black wire from the octagon box to the wire nut the black wire from the LINE of the GFCI went, and your white wire from the octagonal box to wire nut the white wire went from your LINE of the GFCI.

And don't worry about the red wires for now....