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Step 1: Supplies & Equipment
100% wool is the best yarn to use. Feel free to experiment with other natural fibers. Synthetic fibers generally don't take dye very well.
You can use any powdered drink mix, but make sure it's unsweetened powdered drink mix. When you're buying Kool-Aid, look at the color of the glass Kool-Aid Man is holding - not the Man himself or the packet - to know what color the drink mix will be. A reference photo of yarns dyed with Kool-Aid is available from Knitty.com, here.
Most guidelines on dyeing with Kool-Aid will recommend 1 packet of Kool-Aid for every ounce of yarn. I usually use a lot more - here I have 3 packets of lemonade (yellow), and 4 of lemon-lime (green), for 4 ounces of yarn. This is overkill, by a little bit - but if you want vivid colors, you need a lot of Kool-Aid.
You'll also need a saucepan, a bowl, and a big spoon.
Step 2: Make a yarn ball & get it wet
Soak the ball in water. Squeeze it gently, if you need to - try to saturate it as completely as possible.