Copying songs on to iTunes
If you want to play music on your iPhone using the Music.app and you haven't purchased the music on the iTunes Music Store (iTMS), you have to sync your iPhone and the music with iTunes first.

Play music with the Music.app

You can play music with the Music.app if one of the following conditions are true:


You have purchased the song on the iTMS
you have subscribed to iTunes Match and your preferred song is available
you have added the song to your iTunes library on your computer and synced your iPhone with the same iTunes library (enable "sync music")
you have subscribed to iTunes Match, added the song in iTunes on your computer and turned on iTunes Match on your iPhone


Please let me know if this answers your question.



edit:

Alternative method

Play music on iPhones/iOS with alternative music players

You can download and play purchased music stored in your Dropbox if the following conditions are true:


The songs have no DRM applied (encrypted audio containers like "m4p")
you have downloaded an alternative music player from the App Store
you have already downloaded the song or a data connection to download it