If you don't mind cheating a bit this achievement is super easy to gain without even singing the song at all. All you need is access to Expert mode as this one provides around 30% boost on the points you get and you will need a fan. Similar to the beginners singing in Guitar Hero this game detects the wind done by the fan as singing. So put your mic in front of the fan, start the game and get the achievement without singing at all. Sometimes you have to play around where to place the mic to get the best results.
Highest score I was able to get with a fan was around 116.000 Points on "Problem" from Ariana Grande feat. Iggy Azalena. But there are enough other song where you can get over 100.000 points. I didn't manage to get also the 120.000 points achievement but several other achievements are also possible by just having the mic in front of the fan.