Allow the chicken to sit in the cilantro lime marinade for at least 30 minutes before grilling. Preheat the grill over medium heat. Place the chicken on the grill to cook. Grill for about 6 minutes on each side or until no longer pink. While the chicken is cooking, add the tomato, avocado and lemon or lime juice to a small bowl, tossing to combine. Once the chicken is finished cooking, remove from the grill and transfer to a clean plate. Place a slice of mozzarella cheese on top of each chicken breast and top with the avocado and tomato mixture. Top the chicken with the chopped basil and serve.