You need tissue paper that bleeds well, I got ours at Michaels and it was called "Bleeding tissue paper" but not all colors bled well, if at all, so you need to experiment.
Pour some water and a little bit of vinegar into several small jars. Use one jar for each color you are using. For best results use similar colors for one egg, for example blue-purple-red, or light green-yellow, etc. We used red-orange-yellow. Tear (and let your child do it too) small pieces of tissue paper, dip them in water and put on the egg (we used blown eggs and I had to make sure Kitten does not squeeze too hard, the girl loves eggs!).
(As you can see, my pieces were too big)Our orange did not bleed at all, so I dyed it with orange food coloring, which eliminated all white spots and did not change red and yellow, which I was very surprised and pleased about. The result is not fantastic beauty-wise but Kitten enjoyed the process very much, which was my goal. I will make the pretty ones when she is asleep.