You simply decide it's in your own best interest to have him leave and make that as important to you as you would any other big life decision. I don't know if this guy is a boyfriend or husband--if you're married to him, or if you live together and he simply REFUSES (as opposed to talking you out of it) to leave, things are a bit more complicated from a legal standpoint.
When I kicked out the last guy I lived with (not an alcoholic, but other issues), I simply told him (after he'd dragged his feet for weeks on getting out) that he was leaving TOMORROW and I was staying home from work to make sure he did it. He started complaining, "But I have no place to GO..." and I said, "Not my problem." He wound up leaving and staying at a motel for a few days till he could get an apartment. When you're done, you're done.