How to upload songs to Google Play Music using the Chrome browser
Google has final given Google Play Music user a way to upload their songs without having to install Music Manager on the desktop.

To enable Google Play Music for Chrome, click the gear icon in the upper-right corner and select Labs. Enable Google Play Music for Chrome, then click on "Save changes."
Once it's enabled, go to My Library and click on the orange "Add music" button to start the Google Play Music for Chrome setup.
After you've completed the setup, your music collection will begin uploading. The next time you need to upload songs, you'll only need to click on the "Add music" button from My Library.
While your music is uploading, you can also try out the new mini player by clicking on the pop-out arrow, located all the way on the right side of the player controls.