Prepare the Chiles: In a dry skillet over medium heat, toast the guajillo and ancho chilies for about 1-2 minutes until fragrant. Remove them from the heat and place them in a bowl. Cover them with hot water and let them soak for about 15-20 minutes until softened.
Blend the Sauce: Drain the soaked chiles and transfer them to a blender. Add the onion, garlic, cumin, oregano, apple cider vinegar, and about 1 cup of beef broth. Blend until smooth and set aside.
Season the Meat: Season the chunks of beef chuck roast with salt and pepper. Place the meat into the slow cooker.
Combine Ingredients: Pour the chile sauce over the beef in the slow cooker. Add the remaining beef broth, making sure the meat is mostly submerged.
Cook: Cover and cook on low for 8 hours or high for 4-5 hours, until the meat is tender and easily shreds with a fork.
Shred the Meat: Once cooked, remove the beef from the slow cooker and shred it using two forks. Set aside a cup of the cooking liquid to use as a dipping sauce.
Assemble Tacos: Heat the corn tortillas on a skillet until warm and pliable. Fill each tortilla with shredded birria meat and top with chopped cilantro and diced onions.
Serve: Serve the tacos with the reserved broth on the side for dipping. Squeeze fresh lime juice over the tacos just before eating for an extra burst of flavor.