Install Steam on an external drive?

Steam will work on an external drive, and will run perfectly fine as long as you're not trying to simultaneously run it from the computer's internal drive.

Note: Even though installing on an external drive means the games / client themselves are on the external drive, you may wind up with your saves or DLC on the computer, rather than the external drive. (Exact effect depends on the game.)

See also: Is it possible to install Steam games to two different locations?

Steam now has inbuilt support for creating new "Steam Libraries".

You can also manage your Libraries from Steam → Settings → Download & Cloud → Steam LIbrary Folders.