Wash basil and pat dry. Drain pineapple juice from can, reserving the chunks for the skewers. (You will only use about half of the chunks, so go ahead and save the other half for another use.)
Put basil, pineapple juice, olive oil, salt, and pepper into a food processor or blender. Process until basil is thoroughly chopped and integrated through liquid. Pour into a large bowl or shallow baking dish. If you want to reserve some of the marinade in a separate bowl for basting, do it now.
Cut chicken into chunks approximately 1 inch by 1 inch. Add to marinade and toss with tongs to coat. Cover with plastic wrap and put in refrigerator for at least an hour.
When ready to cook, put bamboo skewers into a shallow container of water and let soak. Chop onion and pepper into large pieces.
Remove skewers from water and thread chicken, onion, pineapple, and peppers onto the skewers in an alternating pattern. Place finished skewers on a large plate, and repeat until you’re out of chicken.
Grill over a medium-hot flame for 15 minutes or until an instant-read thermometer in the chicken registers 165°F.