It sounds like you have a charging system problem and maybe a defective alternator charge light or the circuitry that drives it. For a simple check, measure the voltage on the battery posts (not the cables or clamps, the posts) while the car is running at fast idle. You should see around 14.5 volts if the alternator-regulator is working properly; if you see only ~12 volts, either the alternator or its (possibly internal) regulator-rectifier are probably bad. It is also conceivable that a fuse to the alternator is blown.
It "might" be the case that what is causing it not to charge (if that is what is happening) is also draining the battery through defective regulator-rectifier circuitry.
It "might" be the case that what is causing it not to charge (if that is what is happening) is also draining the battery through defective regulator-rectifier circuitry.