How do I install Diablo 3 on an external hard drive without admin privileges?

There is no possibility to install Diablo III without admin privileges. You have to connect your external hard drive to another machine where you have the required privileges and install the game from there.

Afterwards you can switch back to the machine where you have no admin privileges and play the game from the external hard drive you installed it to. You don't need to complete the installation procedure on the machine where you want to play the game.

This also works with Starcraft II and World of Warcraft.

You cannot install the client to your external hard drive from a PC where you do not have administrative privileges. But you can instead copy an existent installation to your drive and the launch Diablo 3 from there.

Pay attention that Diablo 3 and Battle.Net needs various ports opened and if you are going to play behind a firewall they will be likely closed. Is it also possible that you cannot even start the game if the PC you are going to use has been "configured" in a strict way.

While you can copy your Diablo 3 folder to an external hard disk, it is worth noting that Diablo 3 stores additional data in the following locations;

C:\ProgramData\Battle.net
C:\ProgramData\Blizzard Entertainment

These folders contain the Battle.Net Agent in addition to Diablo 3 Launcher updates and will be re-created as you launch the game.

I have not tested if simply running 'Diablo3.exe' directly from the game folder will allow the game to run without the Launcher, but if it does the game will need manually updating for each patch.