It's tough to fight clinical depression alone, much less alcoholism at the same time. Remember, though, that alcohol is in itself a depressant. Drinking isn't going to make you feel better, only worse. And alcohol will have an effect on the success of the antidepressants also. Thus it's imperative that your doctor knows you're an alcoholic.
Don't give up on your recovery. It isn't easy. One thing you hear a lot about in the program is the concept of a higher power. Alone we can't do it, so we ask for the help of a higher power. When that finally sank through my thick skull I found the going to be much easier. You may want to try it.