How can I add more system apps to a custom ROM? I have an android ROM that I can flash onto the phone using a tool like smart phone flash tools, but what I want to be able to do is to modify the system image to add APKs in the system/app directory and then repack and flash the ROM. I want these newly added APKs to behave as part of the system.

I have looked at a couple of tutorials for unpacking and repacking it, but once I repack it, it becomes unusable. Is this the correct approach or am I missing something? I'm not sure what I could be doing wrong.
Most of the time when a unpacked and modified ROM gets repacked, it will become unusable/unstable because it failed to get resigned with the proper key.

You need to have the zip package resigned from the original firmware key. That is why you are experiencing those hiccups. Also if any of the apps depend on libs and/or resouces, you would need to make sure they are in place as well - prior to repacking and resigning

For a decent write up take a look at this: http://androiddomain.weebly.com/build-your-own-rom.html. This will show one of the proper ways of repackaging a ROM.