In a medium bowl, stir together the garlic, tomatoes, vinegar, olive oil, and ¼ teaspoon salt.
Add 2 cups of the tomato mixture to a 6-quart slow cooker. Sprinkle both sides of the chicken with the remaining ¼ teaspoon salt, then add to the slow cooker. Top with an additional 1 cup of the tomato mixture. Set aside the remaining tomato mixture, covered, for serving.
Cover and cook on high for 4 hours or low for 6, or until the chicken is cooked through and shreds easily with a fork.
Remove the chicken to a cutting board or plate, shred with two forks, then return to the slow cooker, stirring to combine with the tomatoes and juices.
Serve the chicken, topped with the remaining fresh tomato mixture and sprinkled with basil.