I have a 2nd 12v car battery connected to power a 1500W inverter. How will that affect the alternator charging system?

Do you have the 2 batteries in the car connected? if so, how?

The charging system in a car is not going to be aided by a second battery if you are running the inverter while the engine is running. Since the batteries will only be used if the engine is not running, or is the amount of power being used is greater than the amount of power provided by the alternator. If you were pulling more power than the alternator can supply, then the check engine light is come on. If you are using the inverter while the engine is running, you are wasting the second battery.

If you are running the inverter while the engine is off, then it will drain the 2 batteries, and then when you start the car the alternator will charge the batteries back up. The advantage of the 2 batteries is that the inverter will run longer before completely draining the batteries. Basically, if you have 2 batteries of the same power and you drain them half way, it would be around the same as if you drained one battery completely (from the charging standpoint). Standard alternators are not built to keep charging up your battery, or multiple batteries. If you are planning on running the inverter running off the batteries on a frequent basis, you should look into having your alternator upgraded. This is common in vehicles that need extra charging power, such as in cars with large amplifiers. I take my alternators to a place that rebuilds them (they also rebuild starters, electric motors, etc.). They always seem to have the ability to bump the power output, last time it only cost around $45

Ultimately, the extra 1500Watts comes out of the alternator, no matter how many extra batteries you have. It will put a significant load on the alternator and at the very least, will decrease it's life.