I would be more likely to believe mental illness is a causal factor than personality disorders. Except I'm pretty sure the research has shown that the relationship between the two is that people with mental illnesses are simply more vulnerable to abuse alcohol. But then that puts us right back to the question of whether or not alcoholism is a disease.
Personality disorders are also a subgroup of mental illness. All these things (alcoholism, personality disorders, depression, bipolar, schizophrenia) are under the umbrella of Mental illness. Like acne is part of the category dermatology.