It wont just play the audio out of your psp's music directory, the mp3s need to be converted to a different format and placed in a directory within your <drive letter>\PSP\SAVEDATA folder on psp.
1) Download the codemasters custom tracks from
HERE (the rockstar one doesnt allow you to convert from mp3s, only from cd). I couldnt attach a copy to this post due to this sites insanely small file size limit.
2) Install and run it, from the 'EAC' dropdown menu in the top left, choose 'convert mp3 music to harddisk', select the mp3s you wish to convert (can select multiple files at once), click open, choose a place to store the output files temporarily and click save.
3) Once its done its work, browse to where the output files are and change the file extension of all the files from .toc to .gta
4) Plug your usb cable into your psp, browse to the folder ../PSP/SAVEDATA and create a folder inside it named as below:
EUROPE - ULES00502CUSTOMTRACKS
US - ULUS10160CUSTOMTRACKS
(you should already have a folder with the same name, minus the CUSTOMTRACKS at the end, which is where the save data for your copy of gta is stored)
5) Copy all your files (now renamed with a .gta file extension) to this new directory.
6)Play GTA: VCS, pause, go to audio options and enable custom tracks, get in a car and enjoy.