1999 Hyundai Sonata car trouble

It was really snowy out today and I was attempting to climb a hill the rpms got really high and then my brake light and my battery light came on. Very shortly after that the car got very hot and I had to park it. When I got back and started it the brake and battery light were still on I drove to the gas station and turned the car off and refilled the coolent that had came out earlier when it got really hot. When I tried to start the car the battery was dead. I got a jump and the car started right up I drove home and when I got home I was trying to let it run to charge the battery and It the power was getting very low as well as the rpms were jumping from between 300 to 2000 then when I placed the car in reverse It died and the battery is completly dead again. It seems like it could be an alternator problem but I am confused as to why It was overheating also.