Although this music isn't 100% the same as the PS1's music, it's pretty much the best available without a complete overhaul of the game.
Note, I did not create the modifications for this. I merely wrote the guide. All credit for the files go to
Samuel Slight
.

Step 1: Downloading the files

First, download
[dmusic.zip]
and save it somewhere on your PC.
Once you have downloaded this file, extract them to a folder names /dmusic/

Step 2: Replacing the original music

Navigate to \Steam\steamapps\common\FINAL FANTASY VIII\Data\Music\. You will see 2 folders and 1 file.
/dmusic/
/stream/
eyes_on_me.wav
Now stream and eyes_on_me.wav are both full quality audio, similar to the PS1. However, /dmusic/ is low-quality sgt files. sgt are similar to midi in that they use soundfonds to create audio, not the ideal option.

First, create a copy of /dmusic/ and name it /dmusic_old/. This is incase you wish to restore the original music tracks, or you mess up the installation.
Once you have done this, copy the /dmusic/ folder that you extracted from dmusic.zip earlier to \Steam\steamapps\common\FINAL FANTASY VIII\Data\Music\. It will ask you to overwrite files, press "Yes to all". By doing this it ensures that any missing files still remain.

Once this is done, you should have a structure like this:
/dmusic/
/dmusic_old/
/stream/
eyes_on_me.wav
Now if you load FF8 through Steam, you should have the new, enhanced music! Enjoy!