This can be completed at the start of Chapter 4. Once you reach the camp, there will be a microphone somewhat center in the area. Use this by clicking on it and Norah will comment that there is no power. Keep clicking and she will eventually sing into the microphone. Just mash the click, took between 10-15 clicks for me
Note: This must be done prior to turning the power on, as the achievement is for specifically making her sing into a dead microphone.