I keep a variety of chile pepper plants indoors over winter and deal with this constantly. Here's what I've learned about aphids. They will kill your plant if you aren't vigilant about them. A neem oil spray can help but won't solve the problem entirely. Some formulations of neem oil are harder on the plants than others so test your solution every time you get a new bottle. If you have a handheld shower wand, it is worthwhile to put your plants in the shower about once a month and blast off the little buggers. And regularly fertilizing your plants will keep them strong enough to resist the attack. The best method I have seen, is to use water from a fish tank to water the plants. Good luck!