Preheat an outdoor grill for medium heat and lightly oil the grate. Cut four 12x18-inch sheets of aluminum foil.

Mix ground beef, bread crumbs, milk, and taco seasoning mix together in a medium bowl until well blended. Shape into four 4-inch patties. Mix potatoes and salsa in another medium bowl.

Place 1 patty onto each piece of foil and top with 1/4 of the potato mixture and 1/4 of the Cheddar cheese. Wrap each packet securely using double-fold seals, allowing room for heat expansion.

Place packets, seam-side up, on the preheated grill. Cover the grill and cook, rearranging the packets several times, until patties are cooked through and potatoes are tender, about 25 minutes. An instant-read thermometer inserted into the center should read at least 160 degrees F (71 degrees C).