I would like to make my own food coloring with natural vegetables, what is the technique?
Anything that's hard to get off of your counter can make a good food coloring. Regarding natural sources, remember that you run the risk of also adding additional flavors so take that into consideration. Among other companies, Annie's makes all natural foods including colorful fruit snacks. What they use for the coloring in these snacks are: turmeric, black carrot juice concentrate, and annatto. (I've cooked with turmeric and annatto before and they DEFINITELY stain.) Aside from Google searches, you could probably find practical natural food dyes by looking in the ingredient list from other all-natural foods.

Back to your specific question about beets, this is from eHow:


  1 Put your fresh beets in a glass or enamel pan with just enough water to fully cover them. Bring to a boil and turn the heat down a bit to simmer for 30 minutes. Make sure the water doesn't boil out. Take them off when the beets are not quite done - When a knife goes into them but they're not as soft as you'd like for eating. Remove them from heat.
  
  2 Strain the beets, reserving the beet juice. Peel, slice and chop up your beets. Return them to the beet juice. Soak for four hours.
  
  3 Strain the liquid out and take out 3/4 cup. Add your 2 teaspoons vinegar to this. This is your red dye.
  
  4 Use canned beets and you can skip half of the work. Drain the can reserving 3/4 cup liquid. Add 2 teaspoon of vinegar to this and you're done. You can still eat the beets, too. The fresh ones aren't terribly appealing after this.