Alternator Problems
 

I jump my 944's battery for five minutes before starting. I start it, leave on for 5-10 min, and remove the jumper cable. When I turn the car off and the try to start it again, the battery is almost completely dead (It doesn't even crank).

Before we do anything, what is the condition of the wiring connections. Are they clean and corrosion free? If not, clean them. The ground connections on top of the bell housing are a common source of starting/charging problems.

Any auto parts store will test your alternator, starter or battery for free. They can even test the whole charging system if you manage to get the whole car there. My advice: while you have it (alternator and/or battery) out for testing, go to at least two places for testing. Monkeys work at some of these places. Remember, it's free and you get what you pay for.

Report your findings and we'll go from there.

5-10 mins runnning isn't enough to breath much life back into a battery, even if it is in good condition. The car still runs when you remove the jumper cable right? I would suspect the battery first.