Problem installing outlet off pull chain light fixture

Since your light is on a different circuit than the outlet, you'll need to run a switch loop from the light to the 2-gang box using 14/3 (yes 3) cable, with black as the hot, red as the switched hot, white as a spare neutral, and the bare or green wire as well, the ground (aka EGC) :)

In the light box, you'll take the black that currently goes to the fixture hot and connect it to the black of the switch loop instead; the fixture hot then gets wired to the red wire of the switch loop. The switch loop white and green wires get wired into the existing white and green wires coming into the box.

After transposing the existing GFCI from the old 1 gang box into the new 2 gang box, you'll want to leave it alone from here on out. To wire up the switch, you connect the brass screws to the black and red wires, the green screw to the EGC, and simply wirenut off (i.e. put a wirenut on the exposed end of) the neutral on the switch loop -- it's there for future use by say a motion sensor or lit switch as per 404.2(C) (neutrals are called 'grounded conductors' in the NEC, btw, if you're a Code newbie and scratching your head at this :):

C) Switches Controlling Lighting Loads. The grounded circuit conductor for the controlled lighting circuit shall be provided at the location where switches control lighting loads that are supplied by a grounded general- purpose branch circuit for other than the following:

(4) Where a switch does not serve a habitable room or bathroom

Electricity doesn't care about color. But electricians (both pros and amateurs) do.

The color is meant to inform both you and any future worker which wires are hot (usually black or red, but occasionally other colors, such as blue), neutral (white or sometimes grey), ground (bare, green or green/yellow striped). If it is not bare, white or green, it is potentially hot.

Code lets you use wire with a different color insulation than the standard if you mark it permanently with the correct color, such as a piece of colored electrical tape, heat shrink tubing or with paint. That way, when you return to the fixture/switch/outlet three years from now, even if you cannot remember what you did, the color of the wires will guide you. And for someone else working on your circuit, it is essential.

Code in all states that I know of in the US require correctly marked wire (either the original jacket or a subsequent marking). Wrong color is a violation. I do not specifically know Michigan code (and it probably varies by local jurisdiction), but it almost certainly follows the standard code when it comes to wiring colors.

On your followup question, it was very common to use 14/2 or 12/2 wire to run a switch loop (a hot line to the switch and a switched hot returning from the switch). The code required that the white wire be marked black (or red) to show it was hot.

Code now requires that all new switches have a neutral, so the old two wire switch loop is not allowed, even if you are using a dumb switch that doesn't need a neutral (some future switch might). So if you are running a new wire, you might as well use 14/3, use the black and red for the switch and leave the white capped in the switch box. You should attach the white at the fixture so that in any subsequent wiring project it will already be properly connected.