In a small bowl combine ¼ cup of lime juice, olive oil, ¼ cup fresh cilantro, cumin, and salt. Whisk until mixed well. Add the chicken and the marinade to a ziplock bag. Marinate chicken in the refrigerator for at least 2 hours.
Preheat an outdoor grill to medium-high heat. Place chicken onto the grill and cook for 4-6 minute per side, or until chicken is fully cooked (the internal temperature has reached 165 degrees F.). Remove from the grill and let sit.
In a small bowl add avocado, ½ cup fresh cilantro, 3 tablespoons lime juice, red wine vinegar, garlic, and red onion. Season with salt. Toss to mix.
Top chicken with avocado salsa.