How do I know If a battery pack is Apple approved

If the battery pack doesn't come with a cable, any reputable manufacturer should be fine. If the battery pack comes with an attached lightning cable (as some very nice ones by mophie do), make sure that you see the Made for iPhone logo.

I've used mophie battery packs and charging cases for years. Excellent quality. No, they're not cheap, though. Going cheap on anything that's related to charging your expensive device is usually a bad idea.

As Meg indicates, Morphie is a good bet. Plus, I also agree, do not go cheap on this accessory. You will regret it.

Do they say "Made for iPhone"? If not, stay far, far away. Beyond that, without knowing the exact make and model, I really can't say.