So, being a non American, I'm not sure I can provide meaningful help here, but the following is what works for me. It is a lot of work, but worth it.
Get 3 pounds of beef brisket, chuck or short rib. Should preferable be sliced into half inch to one inch thick pieces - like steaks. Must have all of the fat and bone still on them. Brown about half of the meat in oil of some kind (I use olive oil) in a pressure cooker. Remove the browned meat when done, remember we dont need to brown all of it.
Then, add roughly chopped carrots, onions and celery to the pressure cooker pot. Fry in the oil and fat (from the browned beef) until the veggies start to soften. Some of it will stick and start to burn - doesn't matter. Continue cooking until the onions start to turn translucent. Then, add all of the beef into the pressure cooker, and add water until the beef is just about covered.
Close the lid of the pressure cooker and cook on high pressure for about 45 minutes. If not using a pressure cooker, cook for 3 hours in a regular pot.
Fish the meat out of the pot using tongs or whatever works. Set aside. Pour the liquid left over through a sieve. Keep the liquid aside. Throw the veggies away. Shred the meat using a fork. Keep any bits of fat (should not be much left over) and discard any bones.
In a deep frying pan, or similar sized pot, fry chopped onion and either chopped celery or chopped fennel (nice for a change sometimes) until both are soft. Add one tin of peeled tomatoes that have been blended using a hand blender (so basically tomato juice now). Add two to four cloves of garlic, grated or crushed. Cook until most of the tomato liquid evaporates. Add the shredded meat back in, as well as all of the stock liquid from earlier. Cook for a further 45 minutes.
Now it is time to make your chili paste, or powder. Use whatever you normally use. Being in South Africa, I don't have the same variety available to me as you guys, so I get by with cayenne pepper, smoked paprika, cumin, black pepper, oregano and sometimes allspice. I also sometimes add roasted red pepper, also blended. Also be sure to add a table spoon or two of worcester sauce, plus a table spoon or either fish sauce or soy sauce. Fish sauce may sound gross for a chilli, but it works well.
During the last few minutes of cooking, add one tin of drained kidney beans.