In a small bowl whisk the marinade ingredients. Place the chicken in a large, resealable plastic bag and pour in the marinade. Press the air out of the bag and seal tightly. Turn the bag to distribute the marinade, and refrigerate while you prepare the rice.
In a medium saucepan over medium heat, bring the rice, broth, and salt to a simmer. Reduce the heat to low and cook, covered, until the rice is tender and the liquid is absorbed, about 20 minutes. Move off the burner and keep covered for 10 minutes. Stir in the cilantro and lime juice. Keep a lid on the rice until ready to use.
While the rice is cooking, prepare the grill for direct cooking over medium heat (350° to 450°F).
Brush the cooking grates clean. Remove the chicken from the bag and discard the marinade. Grill the chicken, smooth (skin) side down first, over direct medium heat, with the lid closed as much as possible, until the meat is firm to the touch and opaque all the way to the center, 8 to 12 minutes, turning once or twice. Transfer the chicken to a cutting board and let rest for 3 to 5 minutes. Cut into ½-inch chunks.
Divide the rice evenly between individual bowls. Top the rice with small mounds of chicken, avocado, tomato, onion, and beans. Sprinkle the queso fresco on top. Serve with lime wedges and tortilla chips.