For the Stock:
Remove the skin and as much meat as possible from a cleaned, whole chicken. Set those aside. In a large stock pot, place the carcass (you can cut it up to fit if you like, but it will also fall apart as it cooks-your call) and enough water to cover and set over medium heat. You don't want it to boil or the fat will get incorporated into the stock. You want a simmer-a gentle one at that. As the scum rises, skim it off adding more water if needed. When you have come up to a gentle simmer, and most of the scum is skimmed (about 20-30 minutes, add your vegetables. I like a quartered onion, carrot scraps, celery, a few stems of parsley, one or two garlic cloves, a teaspoon of black peppercorns, some thyme, a bay leaf, and salt. I save my carrot peelings/trimmings in the freezer for making stock, but if you don't just go ahead and use a new carrot.
Don't put the lid on the pot tightly, or the stock will sour. Leave it to vent and let it simmer (skimming as needed for 4-6 hours). Taste along the way, and when you're satisfied, strain the stock through a strainer lined with a double thickness of cheesecloth (or a clean pillowcase if that's all you have). Let chill, then scrape off any fat that has risen to the surface. At this point, you can return it to the stove and reduce it to glaze, which is nice for cooking, or freeze the stock as is for later use.
Place the skin (it helps to cut them into small-ish pieces) and about 1 teaspoon water in a heavy, small pot. Over medium heat, render until the skins are crisp, and the fat is golden. I know there isn't any point telling you to toss the skins, so sprinkle them with salt, and enjoy. Strain the fat through a fine sieve into a heatproof jar (or measuring cup). Chill.
For the Burrito meat:
Cut your chicken into cubes about 2 in. In a large, heavy frying pan (or stock pot) heat 2 tablespoons fat (I use Crisco for this, but lard, chicken fat, etc. are OK. Regular oil isn't the best, but it works fine in a pinch). Add the chicken and brown over high heat. Remove meat, add a bit more crisco if needed, and add 1 chopped onion, a tablespoon of peppercorns, and the browned chicken. Add enough water just to cover, and bring to a simmer (not boil). Cook, uncovered until chicken is tender (about 1 hour). Strain, reserving liquid. Let chicken cool slightly before shredding apart with your fingers.
Spices are up to you. I like cumin, coriander, chilies, paprika, salt, and epazote. Add 1 tablespoon crisco (yes, more fat-chicken is kinda dry) to pan (use the same one-why do more dishes?) and stir in the spices. Add the shredded chicken and cook about a minute to distribute the spices. Add the liquid back to the pan, and enough water to once again cover the chicken. Over medium heat, cook until most of the liquid has evaporated.
This also works well for beef, and goat filling. Use the filling in burritos, tamales, or a sandwich on a good roll. The cooked, shredded meat freezes really well.