iTunes plays songs one at a time when stored on external drive
Given that you say you're clicking on an album, and iTunes opens, it sounds like you're playing your music from the Finder folder, and not using iTunes itself. If you double-click a song, even if it's within the album's folder, you're just asking for that song.

Once iTunes is open, if you double-click and play a song from iTunes, it'll keep playing the whole album, or whole selection of songs that you've chosen.