Is addiction a disease of the brain?

Yes. Addiction is complicated, and is related to behavior, social factors, heredity, and physiology-->the brain and body. It is unlike a brain tumor, however, in that it does not attack an unsuspecting person behaving in healthful ways. But, after one's behavior has led to addiction, the brain is certainly involved and adversely affected, which decreases one's will power to break the addiction.
YES. The addictive process involves changes in brain structures such as the nucleus accumbens (the reward center) & the locus ceruleus (the fight or flight center that causes some forms of withdrawal). There are other areas involved as well. We can see these changes on functional brain scans & when things improve, we can see those changes too.