There are two separate problems that you have to deal with. The first is the infidelity and any lingering anger you have towards him for that. The second is his alcoholism. Those two problems are related, but they’re still two different problems. The first problem (the moment of cheating) is terrible, but fairly straightforward. You can either forgive or not forgive. But the second problem is the big one.
Your boyfriend cannot control his drinking. He does not have the ability to moderate his intake and he has demonstrated that he can be incredibly hurtful when drunk. You need to seriously consider if you want to deal with this long term. He’s going to either get better or destroy himself, and either one is going to take a long time and be very, very painful for both of you.
There is no shame in acknowledging that this will be hard, and there is no shame in saying that you don’t want to be part of his life during this time. Alcoholism demands more from people than is fair. Some people choose to sacrifice their happiness to be with people who are battling (or not battling) their drinking problems. Some people choose to put their health first and leave. Neither choice is necessarily bad, so you need to make the one that is right for you. I can’t tell you what that is, but I can ask that you think long and hard about what you want to do.
I wish the both of you luck.