With the belt on and what you thing is "tight" can you slip the fan pulley or the alternator pulley by hand? If so, the belt is slipping. Maybe you are at the end of the adjustment slot and just THINK the belt is tight
If you have something wrong in the charging system it might be overcharging, or maybe the battery is simply just that dead
There is not much there to get excited about. You just have to go through everything logically and eliminate one at a time.