[recycled sixtie]

I would start by putting a voltmeter on the battery before starting. Check reading. Then I would start the car and put the voltmeter on with the engine running. If output is greater than before engine started then alternator/generator is doing its job.If no change then alternator is not doing its job. Otherwise there could be a stealth drain on the battery.

The backfiring could be another issue altogether.
Guy