This achievement is cumulative, so it doesn't (and can't) be done all in one day.
Just have the Twitch app running and watching any channel anywhere from 1am to 3:59am (when they say between the hours, they don't mean the starting/ending hours too). Also, this should be going by the clock set on your Xbox, so as long as your Xbox is showing any time during those hours, it should be adding to your progress.
After 5 hours (2 nights) of watching someone play video games, this'll pop.