Let's face it. Discord pretty much sucks when it comes to getting the overlay working. Google is very little help too, but with a bit of trial and error I managed to get it working. It's pretty much a 3 step process.
Go to your User Settings and open the Overlay section. Make sure Enable in-game overlay is turned on. You might also want to set the toggle overlay lock so something different too as the default (SHIFT + `) didn't work for me.
You're then required to manually enable the overlay for each game you want it on. To do this, go to the Games section of your user settings which should look something like the first thumbnail below
Due to all the various versions of Discord and PUBG that have been installed on your system at various times, it's very important that you delete all entries in the list that say PUBG or Player Unknown's Battlegrounds. To do this, simply hover over the game and click the × in the top-right corner.
Then launch PUBG from Steam and close it. This will cause it to reappear in the list of games in Discord, and you can then enable the overlay by clicking the monitor icon.
Finally, go to the Appearance section of the user settings and disable Hardware Acceleration.
Discord will ask you if you're happy to restart the application. Hit Okay and you should be good to go.
At this point, be sure to go back and confirm the overlay hasn't disabled itself on either the game or on the overlay page itself.
Launch the game, hit your hotkey, and you should see the overlay options come up
ALT + TAB out of your game, and make sure the overlay is enabled in User settings > OverlaySHIFT + \