Yes, you should be able to do most of your development with files stored locally on an external drive. You would work with these files the same way you would work with other local files, you would just need to figure out the correct file path to use to view your files in the browser.
For example, on my Mac, here's what the path to a file on an external drive looks like:
The easiest way to figure out what path this should be is to open a file that's on your external drive in a web browser.
Other than that, I think that this would be pretty much the same as developing your website locally.
Hope that helps!