No simple solution. While there are many different products that make these claims, there is no one cure for all conditions. Popular non-prescription ingredients include salicylic acid, benzoyl peroxide, and retinol preparations. However these can be very drying ad make some conditions worse if not used in combination with appropriate exfoliation and moisturizers.
Diet connections? From a holistic perspective, food sensitivities may sometimes be involved (different from allergy). Common foods i generally recommend avoiding are grains (particularly glutenous grains), all dairy products and refined sugars. Look into the paleolithic style of eating as an option. Consider seeing a clinical nutritionist or a holistic doc for further evaluation.
Get real Rx. "pop" remedies are at best adjuncts to scientific treatment. Tea tree oil may help but has given teenaged boys breasts. If topical benzoyl peroxide has failed, a physician's office visit is in order. Acne is a serious, chronic disease that can do grave physical and emotional damage. Don't let your reluctance to use 21st century medicine cause you to neglect the health of a teen. Best wishes.