Battery keeps dying,starts immedetily when you hook jumper cables to it,could this be the alternator?

Very possible. Charge the battery and have it load tested, then put a volt meter across the battery with the engine running. Voltage shown should be above the volts shown on the meter before you started the engine. Normal reading should be around 13.5 and should not drop to less than 12.5 with all accessories turned on (including high beams). Voltage should not continually drop over time either. A steady high voltage reading (14 volts or higher) is not acceptable. . High charge rate indicates either a full field condition in the regulator, a shorted field wire or a bad battery.
Always check belt tension before testing as a slipping belt can alter charging rate.

It could well be that if its not charging the car battery up,,or it could be you have a dud battery that just wont hold its charge any more