For those that don't have any money, or don't want to spend any money for this achievement, there's multiple track packs/singles that can be downloaded for free:
Ferret/Metal Blade Track Pack - 3 songs
Guitar Duel Track Pack - 2 songs
National Anthems Pack - 5 songs
Neversoft Thanks the Fans Pack - 6 songs
Neversoft Track Pack - 2 songs
Reggae Track Pack - 3 songs
"Anything" by An Endless Sporadic
Download all of those, and you should be right at 115 songs. Just update the library after everything is downloaded, go to the menu of songs in Quickplay+, and it should pop. There's probably a couple of other free songs, but I can't remember them.