Mental illness causes the prefrontal cortexes (the part of the brain that is responsible for executive function) to function at a lower level causing irrational thought. Anxiety, for example, causes an imbalance in the brain that correlates with the Prefrontal Cortex, which controls impulse, executive decisions, planning, and moderating social behavior.
The individual coping with anxiety will be in a state of dysphoria and find a means to rid themselves of that state.
This can lead one to self-medication with a variety of substances. However, an addiction can easily develop when someone self-medicates for their mental illness symptoms. The most common withdrawal effect among drugs and alcohol is anxiety. Once the drugs or alcohol wears off, the individual can have even worse anxiety leading them to use even more to numb the feeling of anxiety.
Sadly, this can quickly become a vicious cycle and lead the two to feed off each other. When the substance fades anxiety goes up and when anxiety goes up the craving for the drug goes up.