How do I change background color of text messages?

In addition to AppleAnswerDude's excellent explanation of why you're seeing a green background, I would add that you can't change the color. Green is for SMS/RCS and blue is for iMessages.

You can certainly let Apple know that you aren't a fan here:

Product Feedback - Apple

If you're seeing green message bubbles it means that your message was sent using MMS/SMS instead of iMessage.

There are a few reasons why the bubbles are green instead of blue. (These are the only two colors for messages sent on your iPhone)

1. The person that you sent the message to doesn't have an Apple device. (Android / Google phone users)
2. iMessage is turned off on your device or on your recipient’s device.
3. iMessage is temporarily unavailable on your device or on your recipient’s device.
4. You might need to update your settings for Messages after you set up a new device.