I have a laptop that the internal HDD is dead in. My question is how would I install Ubuntu on a new external HDD and have the laptop boot from it without an internal drive being installed. I've looked at other questions and while some are similar, I havent found anything that exactly matches. I mainly want to use Ubuntu to run Wine so I can play World of Warcraft from a private (legal) server. Thank you in advance.
The short answer is: It is no problem to install Ubuntu to external HDD using the standard installation process.
There are several things you should take in to account here before you do.
How have you determined your HDD to be dead? In other words, are you sure it's irrecoverably broken?
What kind of external HDD are you using? Is it USB 2.0 or 3.0? Does your laptop have USB 3.0 ports?
Would you be comfortable opening your laptop to replace or at least remove the broken HDD?
Have you tried running Ubuntu from a live USB stick and tried to access the files on the broken drive? Have you tried some diagnostic tools like the SMART data & self tests of the Ubuntu disk utility when running from live USB. It can be found by typing
Disks in the dash.
USB will not give you the same data throughput rates as the internal HDDs SATA connection. Especially when your drive or ports are only USB 2.0 it may be a very bad experience especially when gaming.
Given you have the means to afford a new hard drive and you or someone in your circle of friends or family would be comfortable with replacing it in your laptop I think this would be the most satisfying and long-term viable option. Keep in mind that you would most likely need a 2.5 inch HDD if you buy a new one. You can get new ones with 500GB starting at 41,90€ online.
If you've thought about the 3 points above and think that installing to external HDD is the most viable option for you at the moment, it should pose no problem for the standard Ubuntu installer.
If you have disconnected the broken hard drive and your external drive is the only HDD present you can just tick the 'Erase disk and install Ubuntu' option in the installer.
If the broken HDD is still connected and can be recognized by Ubuntu you have to pick the 'Something else' option from the installer. Then you just select your external HDD for bootloader installation from the dropdown menu seen in the image (note: your external HDD will not be /dev/sda. See which descriptions matches your drive)
Delete any existing partitions on it (everything under /dev/sdX where X is the letter of the drive you've selected for bootloader installation) and add a new partition on it consuming all unallocated space.
Then just proceed normally with the installation.