I would definitely suspect the alternator then.
I would recommend picking up a voltmeter (preferably digital).
Measure the voltage with the engine off. It should be 12.5 volts, hopefully a bit more. If it's below 12 the battery is mostly discharged, and should be put on a charger.
Once the engine is running, check the voltage across the battery terminals. It should be closer to 14 volts. If it's less than about 13.6, then I would suspect there is an issue with the alternator or the PCM (less likely).
Also, go back and clean and tighten every electrical connection, especially the wires connecting to the battery and to the alternator. If you can locate where the ground strap on the battery grounds to, clean that too. If there is even a little corrosion or dirt in the connections, it can cause some odd issues.
Take some steel wool and make every connection shiny. If there is corrosion buildup on a battery terminal, use a mixture of baking soda and water and scrub it away