Place peppercorns, dill, parsley, bay leaves, leeks, carrots, celery, chicken, wings, and backs into a large stockpot. Add stock and 6 cups cold water. Bring to a boil. Reduce to a very gentle simmer, and cook for 45 minutes. Liquid should just bubble up to the surface. A skin will form on the surface of the liquid; skim this off with a slotted spoon, and discard. Repeat as needed. After 45 minutes, remove chicken from the pot, and set aside until it is cool enough to handle.
Remove the meat from the bones, set the meat aside, and return the bones to the pot. Shred the chicken, and set aside in the refrigerator until ready to use. Continue to simmer the stock, on the lowest heat possible, for 3 hours, skimming as needed. The chicken bones will begin to disintegrate. Strain the stock through a fine sieve into a very large bowl. Discard the solids. Place the bowl in an ice bath, and let cool to room temperature. Transfer to airtight containers. Stock may be refrigerated for three days or frozen for four months. Refrigerate for at least eight hours, or overnight. If storing, leave fat layer intact to the seal the stock. Before using, remove the layer of fat that has collected on the surface.