Alcoholism and mental illness can be genetic, as far as I know, but it's not a definite thing. If you're parents were both addicts, it doesn't determine your fate that you will be one as well.
My father is an alcoholic, but he is the only one that is from his side of the family. My grandpa and mother are probably alcoholics too, but my mother probably turned to alcohol due to mental issues as she's bipolar.
Mental illness also runs rampant in my family and while I'm no alcoholic, I have been treated for minor anxiety disorders in the past.