How can I get multiple computers to use my iTunes library on an external hard drive?
You're missing the "Library". The media folder contains the actual audio files, but you need a library pointing to it. The 1st approach is, move everything and make all the iTunes open the same library. 

Won't work because when Machine 1 opens the Library.xml, it locks it. Bummer. But turns out that there's a way to do it, if you're willing to involve Dropbox and some patience, check the link but for reference, the idea is:


Move your entire iTunes folder to your home network drive.
Copy the iTunes library (database file, XML files) to your Dropbox folder
Create an alias of the network drive’s iTunes Music folder and place it in your Dropbox folder. Be sure to rename it to iTunes Music (remove the word “alias”). Create an alias of the Album Artwork folder, too. If you skip this step, whenever you import music to your library, it will be copied to your Dropbox folder instead of your network drive. The aliases ensure iTunes places all new library items on the network drive.
On each Mac, configure iTunes to use the library file located in the Dropbox folder. Do this by holding option on your keyboard when clicking the iTunes icon in your dock. You will be prompted to choose an existing library or create a new one. Click “Choose Library” then navigate to your iTunes database file in your Dropbox folder.


You can now access your iTunes library on each Mac without using Home Sharing. iTunes will behave identically on each Mac; you can add music, edit playlists, and sync your iDevices without Apple’s usual restrictions.

There are a couple of Caveats and Gotchas, so check the link for more information.

Or you could just do what I did… get iTunes Match and enjoy a simple easy to use environment. I am really happy the way it works, the caveat is, of course, you will be using bandwidth (or needing it) to stream/download a song. So if you have no connectivity, you can't listen (you can always have one or more machines with a copy of the songs, I have my "master" machine with the 90GBs of music locally, but the Macbook Air and iPhone 'stream' on demand). 
The other caveat is that if you "save" local tracks (you can), then you're effectively duplicating your copy. But hey, you can remove/download as you please, as long as it's in the "cloud".

Finally, iTunes Match is for music, so for videos and such, you will have to use Home Sharing (with the annoyance that it's "read only" for the connecting computers), but for music, you can buy/add from any authorized device… and you need to be in a country that supports it (and willing to pay 25 dollars/year). :p