You can install to the drive you boot from and it is not too difficult, see:
Can Ubuntu be installed to the pendrive it was booted from?
It is easier if you buy a flash drive of at least 2GB, (cost should not be more than a buck or two), and make a Live installer using UNetbootin or Rufus
Then you can do a Full install to the external hard drive.
Best to unplug your internal drive before proceeding.
Use "Something else" when partitioning.
If you want your Windows internal drive to be able to see space on the external, leave your first partition as NTFS.
After installation plug in your internal drive, boot the external drive and then update-grub, this should add the internal drive to the external drives grub menu.
In BIOS or UEFI make the external dive the first to boot, now if the external drive is plugged in you get a choice to boot Linux or Windows.
If the external is not plugged Windows will boot.
There is lots of info on doing a Full install to USB at:
How do I install Ubuntu to a USB key? (without using Startup Disk Creator)
One method on that page that might suit you is installing to USB from VBox, no Live USB is required, in brief:
Create a new virtual machine using your Linux choice as the ISO.
When you get to Try/Install choose Try.
Mount your external drive.
Now choose install.
Use something else if you want a Linux/Windows NTFS or FAT32 partition.
Update-grub the first chance you get.