The process for installing to an external is pretty much the same as installing on an internal. You just gotta boot via a USB drive or DVD with the Ubuntu ISO loaded on there. During the install you just select your external and install on there. Super simple.
And, it's best to Disconnect your Main Internal Drive so you don't Install to it by accident and overwrite windows 10!
You're best off buying an inexpensive USB, I know you don't want to hear that but that is the way it is.
A DVD just don't cut it for me anymore, things are way to slow. If you get a small inexpensive USB you will be able to run from that just fine without Installing. You will be creating a LinuxLiveUSB, so some use the Term "Install", but it is not a full install and only takes a few minutes to create. ISO Download will take you a while depending on your download speeds, then again, the USB Creation will take about 7-8 minutes by experienced folks.
You could also do this with your External Drive I guess, but that seems like a waste of space, because it should be a dedicated drive to keep things simple, this is why the USB route is best IMO. I've used the Lexar Twist drives with success. There are a few drives that don't like to boot with Linux on them, I don't keep track of them all but can say some of them are Sandisk brand. Most will boot with Linux IMO, and they don't need to be very big, 4GB will do, even smaller maybe.
The other option you have to try it out is a Virtual Machine inside of Windows 10.