How do I move my iTunes library on an external hard drive to play off another computer? I want the iTunes to stay on the external hard drive
Within iTunes Preferences, go to the "Advanced" tab and you'll be able to change the "iTunes media folder location" - just provide the path for your external hard drive.
Bear in mind that you'll have to make sure that any machine accessing the library is authorised to play the content (found in the "store" menu).
If you've already got iTunes using a library on an internal drive, you'll need to manually copy the files to the new location and you might need to import the data again (through "File" -> "Add to library") before it appears.