I am using Linux Mint 20.3 Cinnamon 64-bit on an old Alienware laptop with keyboard and mouse. In my situation I already owned and subscribed to ESO, but had not purchased it through Steam. Therefore I had to first buy the Steam version, and then link my Steam account to my ESO account. The steps I used are summarized below after collecting the info from different forums and sources.
1. Start Steam, click on the drop down menu "Steam", and select "Steam Play". Check the box for "Enable Steam Play for all other titles".
2. Buy the Steam version of ESO if you don't own it already.
3. Check that your game downloads are set to the correct drive. You do this by going to the main Steam window's drop down menu called "Steam" in the upper left corner. Then go to Downloads, and click on the box titled "STEAM LIBRARY FOLDERS". If you only have one hard drive, you should see it displayed as a Local Drive and how much GB are used out of the total available for it. However, if you have an SSD for booting, and a separate hard drive for installing games you will need to add another game location by selecting the plus sign next to the Local Drive box and pointing to the drive and directory you want Steam to use for your game downloads and installations. This assumes your other drive is already setup and working within Linux of course.
4. Install ESO from Steam
5. Go to your Steam library of games, right click on the ESO game, and pull up the Properties.
a. Under the General section set the Launch Options to the following:
PROTON_SET_GAME_DRIVE=1 %command%
b. Under the Compatibility section make sure the "Force the use of a specific Steam Play compatibility tool" checkbox is checked, and change the selection below it to Proton 6.3-8.
6. To get my addons for the game, I simply used a flash drive and copied the "Addon" folder from an installation on my Windows computer as well as the "Saved Variables" folder and copied them to the Linux computer under the "Live" directory which for me was located here:
/home/<username>/.steam/debian-installation/steamapps/compatdata/306130/pfx/drive_c/users/steamuser/My Documents/Elder Scrolls Online/live/
The 306130 number is the number that Steam uses to identify the ESO game. You can find these numbers by using your web browser to find the game on the steam website. The url will have that number at the end of the address, e.g.
https://store.steampowered.com/agecheck/app/306130/
7. Play the game!! I did all my crafting writs on 16 toons without issue using Dogulbon's Lazy Writ Crafter and all my other normal addons active as well.
fyi I went through several failed attempts trying to use Wine, and/or Lutris (both standalone and Steam) on PopOS and on Linux Mint. Even after I used the approach above, Steam's default use of Proton Experimental when running the game would get the Launcher started, show the news, update the game, and try to start it, but I would end up with a black screen with the Launcher on top and I could not get out of it except by rebooting. After I set the ESO Steam game to use an older version of Proton (6.3-8), the game seemed to work great.