What simple solution can I apply to clean the cloth head rest on my

try using a small amount of dish soap and a damp cloth.

You can purchase an aerosol can of furniture/carpet cleaner. It's a small amount for spot cleaning. Woolite has been around for years and you can also buy "green/organic" fabric cleaners. Buy and use as directed.

Not knowing what the chair is covered with, I'd suggest Murphy oil soap, following the instructions on it. This stuff is great for removing oily stains, even those annoying ones that happen to cotton knit shirts. Using a soap-and-water solution, 'scrubbing' with a cloth such as microfiber or cotton washcloth has worked well for me on so many things.

If cleaning fails, might I suggest making a small rectangle of complimentary fabric and covering the stain. My husband has off white fabric in his office chairs and I made coverings that could be removed and laundered. And replaced if the decor or his tastes changed.