It’s not a disease in that you might wake up one morning addicted to cigarettes although you’ve never smoked in your life, in the same way as you might wake up one morning with cancerous cells developing somewhere in your body with no apparent cause.
That’s based on a limited definition of “disease”, though. This definition from Merriam-Webster’s Medical dictionary (via dictionary.reference.com) is broader:
an impairment of the normal state of the living animal or plant body or one of its parts that interrupts or modifies the performance of the vital functions, is typically manifested by distinguishing signs and symptoms, and is a response to environmental factors (as malnutrition, industrial hazards, or climate), to specific infective agents (as worms, bacteria, or viruses), to inherent defects of the organism (as genetic anomalies), or to combinations of these factors.
By that definition, mental disorders (including addiction) should be classed as diseases.