Playing through an external hardrive

Solution 1:

Yes you can, the game can be installed to any location and then played from there as long as the entire game is self contained in it's own folder. The game performance may suffer a little because it is being accessed via a data cable which is generally slower than an internal hard drive.

This also helps in terms of moving the game around from computer to computer/hard drive to hard drive.

Solution 2:

It's perfectly possible. Sure, you can play it from any device, but what's more important: The characters, all their items and progress etc. are stored on the Blizzard servers (even for singleplayer).

This means that you can even play the same character at work and from home!