Mix brine ingredients in a cooler--I use a smaller Coleman cooler--and add defrosted turkey (ensure the turkey is fully covered by the brine mixture); let sit for up to 24 hours. Note - The temperature in the cooler should be 33°-40° to prevent bacterial growth. If the temperature begins to get too high, stir ice into the brine until it has dropped back to the desired range.
Remove turkey from brine and pat dry.
Mix Rub ingredients and coat turkey inside, outside, and under the skin. Note - Use your hands to peel the skin away from the body then apply the rub. Use toothpicks to hold the skin down while cooking.
Use a chimney starter to heat some charcoal and place it in the bottom of your smoker.
Pour smoke liquid into the water pan of the smoker and place the cover on the smoker until he internal temperature reaches 250°-300°.
Insert a meat thermometer into the thigh of the turkey, place it on the top rack of the smoker, and cover with the smoker lid.
Add wood chips or chunks to the smoker through the side door.
Let sit, adding more chips every hour (or as needed) to keep the temperature between 250°-300°.
Turkey is done when the internal temperature reaches about 165°. The cooking time should be roughly 20-30 minutes per pound.