If you are decided to install Kali Linux in a different HDD that's a good idea, this will not interfere your Windows 10 anymore and OS stays stable. But if you want things portable, then i have some better suggestions for you.
Rather than using an External HDD get the internal one, and at least use SSD for a faster load times or to prevent crash or hanging problem. Any cheap SSD with the speed of below 350 MB/s, Size of 128GB will be better than a old HDD.
Well Why External HDD? This may create a problem, plugging it in different system, transferring files here and there may cause your OS to be get corrupted. Better use a separate internal SSD, weather you are using a PC or a Laptop.
Things you may need to set up your safe dual boot. If you are using a Laptop.
Benefit of doing this is now you have a Internal Drive with OS that nobody can interfere or misplace. It is separated from you Windows OS and not even windows user can see the other hard drive data (because linux uses ext4 file system).
So in the means of safety and privacy its fits the best.
How to boot after installation. Go to your Laptop or desktop Bios setting, there set your Windows 10 drive 1st boot device. So if you turn on system it will normally open Windows 10. Now if you want to boot Kali Linux then restart system and this time press boot menu key. In the Boot Menu select Linux drive.
Boot Simple and easy like a cake.