It's clearly one of two things, either the battery cannot accept a charge and hence requires replacement or it's not getting charged. I would first start off with a meter and check the voltage at the battery WITH the engine running you want to be seeing 13V +, if so then try applying a load, by say, turning the headlights and blower on, at which point you still should not be seeing less than 13V, the light extinguishing should not be taken as gospel proof the charging system is good.

Martin