Prepare the chicken for grilling. Trim off any fat.
Mix all of the marinade ingredients in a 1-gallon zip-lock bag, then add the chicken breasts to the bag. Massage the marinade into the meat until evenly coated.
Seal the bag and place it in a bowl or on a dish in the refrigerator to protect it from leakage. Let the chicken marinate for at least 2 hours but you can even do overnight.
Next, prepare the salsa. Gently mix peaches, jalapeño, onion, bell pepper, cilantro, garlic, and lime juice, in a bowl. Then season with salt and pepper to taste.
Cover the bowl with plastic wrap and refrigerate for at least 30 minutes. Fold the avocado into the salsa just before serving.
Preheat the grill to medium-high heat and oil the grates. Place the chicken breasts on the grill and cook, covered, for 4 to 5 minutes per side.
Place the grilled chicken breasts on a platter and generously spoon the salsa over the top, allowing the flavors to meld together.
Garnish with a sprinkle of fresh cilantro or a squeeze of lime juice for an extra burst of freshness.