Here is how I went!
And here's how I did it... it was super-easy if a little messy.
As I liked the low-pile look of the old overdyed kilim rugs I began by flipping my rug over to reveal the back. As well as replicating that low-pile look this also exposed the warp of the rug (the cotton threads my rug was woven into).
I then ditched any thought of stripping the dye from my rug (too messy, too many chemicals) and decided to go straight ahead with applying a colour OVER THE TOP of the original design.
1. Mix a custom colour using water based screen printing inks. I used Permaset brand. To make the mixture even and smooth give it a zap with a handheld blender.
2. Pour your colour into a spray bottle and add equal amounts water. Shake with vigour.
Spray an even layer onto the rug, I needed 4 bottle-fulls of ink to completely cover my rug.
3. Allow to dry and fix by ironing the rug with a hot iron. Place a teatowel between the rug and iron to avoid scorching.