To unlock this achievement "Karaoke Superstar", you must complete all eight songs with a score of 90% or above.
As SirDooblay said, four songs will be unlocked through the main story. The next two are unlocked by calling Tiffany and then singing karoke for her. Finally, the last two can be unlocked if you perform the favor "Real Men Don't Karaoke". The person who asks you to carry out this favor for him can be found at the bar inside K-Bar.
Once all the songs are unlocked, get to K-Bar and enter the karoke room. Then start playing each of the songs and go through them, one by one. Your job is to make sure you hit all the notes perfectly or as well as you can. It's OK to make a little mistake here and there, but make sure to keep it steady in the green part of the meter that's located on the top-left corner of your screen. It's easy to hit the notes in the middle, top, or bottom. With the middle, you don't move the analog stick. For the top, push it all the way up. For the bottom, push the analog stick all the way down. But for the spaces that's in between them (up/middle, middle/bottom), you need to touch the analog stick slightly up or down and keep it there until the note is done.
These songs are short and it should take you between two to five minutes to complete each song. When I did this achievement, it took me at least a half hour to get all songs done with a 90% hit rate or higher. The last song I did was "Take On Me" by the Ahas and once I got a 90% percent score on it, the achievement unlocked.
If you are having trouble with this achievement, then I ask you please watch the video I've attached with my solution here. All credit goes to leighbuss for creating and posting the video for all to see and to show how it is done. Just follow the game as they do it and you'll earn it. Keep in mind, though, that all songs play out differently, so you'll have to rely on quick reflexes and timing. Again, it's fine to make a mistake here and there, but not too much!
I hope my solution helps! If you somehow feel to give my solution a negative vote, I kindly ask that you comment in my solution and explain why so that I may correct and adjust it as best as I can. Have fun and good luck!