How do I empty my iCloud pics to an external hard drive?

If you have a Mac computer, you can attach the phone to the computer, then use the Photos application or Image Capture to copy the photos to the computer, and then to the external drive. Another way to do that on a Mac computer is to open Photos with the option/alt key held down and create a new system Library on the external. Once everything is downloaded, reopen Photos with the option/alt key held down, and switch back to the Library on your internal drive. Switch as required/desired.

Article if for iPhoto, but the basics are the same.

iPhoto: Move your iPhoto library to a new location

Photos library - Move your Library to an external storage device

Using the Photos app on a Mac, go to File > Export > Export Unmodified Original. I did this with a specific album of over 1,500 photos on iCloud, created a folder for the photos to go into, and when the download was complete, I moved that folder to an external hard drive.