Translation:His problem is drinking too much alcohol.
There's something about this that seems odd to me. "His problem is drinking too much alcohol" seems a bit like the problem drinks alcohol. Maybe "His problem is that he drinks too much alcohol." would be better. I could be nit-picking here, but it just seems odd. :p
Thats because mentally "drinking" = "alcohol" :) "On ma problem z piciem." - He has the drinking problem. And the oposite but also popular statements would be: "Jego problem to alkohol." or "On ma problem z alkoholem." I would say that this sentence is less awkward if you say: Jego problem to picie zbyt mało alkoholu [too little of alcohol]. e.g. He drinks too litle so it is not in the righ mood :). And the information doesn't look duplicated any more.