Well, it sounds like you have 2 problems:

1) The computer's batter holds a weak charge, and only has a short lifespan.

2) Your PRAM Battery is dead. you can change this pretty easilly, and they cost $30-$60.

Many macs behave erradicly when their PRAM battery's get low.

or, you could check your PRAM battery, hook your leads up to a volt meter, and a good battery will measure between 3.2 and 3.6 Volts.