If you have the required hardware (the mic), this achievement is very easy. Plug it in and go to the Quick Play menu. While you breeze through the songs you notice that you find a "DJ & Vocals" right beneath of "mixed by ...." these are the songs you need to perform vocals on.
Important: You can choose every difficulty you want plus you can totally suck on those vocal performances, as long as you finish them. Start a setlist with the maximum number of songs with vocals (20), put your mic on the table and come back later. Repeat with the next bunch of DJ & Vocals songs.
There are 58 of such songs in total.
I looked it up. Seems you don`t need another controller and sing them all by yourself.
Here are some very helpful suggestions from Darth Trom. Big thanks for that =)
"#1. May be obvious, but if someone doesn't know which songs they are missing, they can usually go to the main menu, click "Rep", click "Achievements" and then scroll down to the Broadway Achievement, and it will tell you there how many more songs you lack, and at least one song that needs to be completed to get the achievement. If you find that you have sung all of the songs on quickplay and the game is telling you that you still are missing songs BUT it won't tell you which songs you are missing.... Check out #2
#2 suggestion.... You might add that if people have not completed each of the bonus setlists gigs on empire mode all of the songs may not be available immediately and achievement won't be attainable until they complete it."
Anyway good luck bros and enjoy this awesome game!
(If you have any questions, feel free to contact me. I'm glad to help (; )
AP E JC said "You should point out that if your controller is turned off in the end of the setlist, you wont gain the achievement. That happened to me. Just play another Vocal mix, and achievement unlocked."
So if that happens to you you know what to do (; Thanks for that information! =)