You need a volt/ohm meter, it will save you hundreds of dollars.
First never use a battery charger on a battery that is hooked up to the vehicle, remove battery cables then charge battery.
The newer electronics can be FUBARed by battery chargers, and even jump starting.
Second, measure battery voltage with key off, set Meter to 20vDC(20 volts DC or just DC volts)
New battery will be 12.8volts
5 year old battery 12.3volts
Now lets test alternator wiring, key still off
Black meter's wire on battery Negative, Or engine metal(ground)
Red meter wire touched to Alternator's B+(big terminal on back side of alternator)
You should have Battery Voltage, whatever you had above should be here.
If not, you have some blown fusible links
If you have Battery Voltage at B+ then unplug the 3 wire connector on alternator
There is a Yellow/white stripe wire, test it the same as above you should have battery voltage.
If not then a fusible link is blown
Now turn on the key
Test light green/red stripe wire, it should have Battery voltage but only with key on, this is the Battery Light wire.
A Fusible link is a short wire that is a smaller gauge than the main wire, it is used as a fuse, it will melt and separate without burning if too many amps are run thru it.
The reason they use fusible links is that they can tolerate a spike in amps without blowing out like a fuse would, and alternators can have amp spikes at first startup.
The splices for the fusible links can also fail, so if you find one of the tests above shows low or no voltage then you will need to open the wiring harness to follow the wire with no or low volts until you find the Fusible link and then check why it isn't passing voltage.
Disconnect battery's Negative cable when tracing the wire.
Read this thread, same year and model as your, and no charging just like yours: '03 4.0 -- low voltage on alternator "A" terminal