Chest hurts when I cough (have a dry cough) and when I breathe deeply. I feel it in my back too. Lung infection?

Pleurisy. Pleurisy is chest pain, often worse with cough or deep inspiration, that results from irritation of the tissues surrounding the lungs. It is most commonly caused by viral infections, especially in young people such as yourself. Other causes include bacterial infections, trauma, autoimmune disorders, tumors and some medications.