Go to the disco club called Maharaja Kamurocho along Shichifuku Street (top, middle part of the map). Pay 2,000 yen to enter the club, and head towards the DJ to initiate the dance mini-game. Complete all five songs at least once to unlock this achievement.
Dancing isn't particularly difficult, but may prove tricky if you aren't good with Quick-Time Events (QTEs). You'd need to move your d-pad to the corresponding squares on the dance floor, and press the correct buttons at the right moment to register a score. To "complete" a dance, you'd need to get at least a two-star score - I got one-star in one of the songs, and it didn't say I "complete" it. After you complete a song, you should see a "complete" mark when selecting it.
You can select the difficulty when you choose the songs. Easy is fine. The songs get harder even on this difficulty.
Midway through dancing solo, you should also unlock a side story for dance challenge. You don't need to fight this guy to unlock this achievement so feel free to skip him for now if you're in a hurry.
Update on points required to pass a song (credits: XF34Rx KiNGzz):
Friday night - 2372 points or more
Queen - 2640 points or more
I'm gonna make her mine - 3030 points or more
I wanna take you home - 3450 points or more
Koi no Disco Queen - 3776 point or more
PS: This can be done any time you're on free roam. I did this in Chapter 2 after the quest to find alcoholic drinks for the five bozos.