I’ve followed the Doom instructions from here.
I put chex.wad in the .iwad folder and created Chex Quest.doom with chex.wad in the first line. I’ve also created a .Chex Quest folder.
I just get a black screen and nothing launches.
I’ve done the same for Doom, Doom 2, Freedoom and Freedoom 2 and they all launch correctly.
Seems to be working fine here. This is my Chex Quest.doom file contents.
parentwad "DOOM.WAD"
wadfile_1 "CHEX.WAD"
wadfile_2 ""
wadfile_3 ""
wadfile_4 ""
wadfile_5 ""
wadfile_6 ""
wadfile_7 ""
wadfile_8 ""
dehfile_1 ""
dehfile_2 ""
Make sure you have a valid Doom and Chex Quest .WAD file within a .Chex Quest folder.
When I format it that way it loads the Chex levels in Doom but with a Doom skin.
What version of Chex Quest are you using? If its Chex Quest 3 I do not beleve that works with PrBoom so you may want to test with Chex Quest 1 or 2
I am using 1 and 2.
They load fine on my PC.
Please provide the contents of your Chex Quest.doom file along with an MD5 checksum of both the DOOM.WAD and CHEX.WAD files you are using. Also add screenshots to show your current folder setup. The more information the better.
You can do a quick MD5 checksum here, MD5 File Checksum - Online Tools
Also as a side note, ensure that your Chex Quest .WAD files are inside your .Chex Quest folder.
If you look up Chex Quest 3 Vanilla you’ll find what you’re looking for! 
Contents of the file:
parentwad “chex.WAD”
wadfile_1 “”
wadfile_2 “”
wadfile_3 “”
wadfile_4 “”
wadfile_5 “”
wadfile_6 “”
wadfile_7 “”
wadfile_8 “”
dehfile_1 “”
dehfile_2 “”
(Chex is an IWAD so it should be the parent WAD)
Chex MD5 hash
25485721882b050afa96a56e5758dd52
Doom MD5 hash
c4fe9fd920207691a9f493668e0a2083
Please see my post here, Chex Quest not launching on RG35XX H - #2 by xonglebongle
And see my post here, Chex Quest not launching on RG35XX H - #3 by razorbeamz
If I format Chex Quest.doom to match what you posted, it loads up the Chex levels in Doom.
I just tested this for myself on my end (like I said I’ve been using the PortMaster port of GZDoom called “Doom Engines”) but I just gave it a crack using RetroArch’s PrBoom core.
I validated against your checksum that you are using the classic original “Chex Quest 1” wad.
While this file “is” an iwad, its actually missing the iwad header, meaning that to play it on a “vanilla accurate” source port, without some extra modding, it has to be loaded as a pwad. So the structure XongleBongle provided is correct, with Doom as the parent wad and chex as the first pwad.
I just tested on my end and can 100% confirm chex quest loads as chexquest and not just as a chex quest skin of doom 1. The only caveat is that you still get a Doom 1 “episode selection” menu but once choosing that, you are actually taken into real chex quest.
If this is not happening on your end then it may be an issue with how you have named your files or structured your files.
Make sure you have a folder in the Doom root named “.IWAD” that has “DOOM.WAD” in it. Then make sure you have a folder in the Doom root named “.chexquest” with “chex.wad” in it. Then finally, your “chexquest.doom” file in the root of Roms/DOOM in the format Xongle provided. Really the folders and files can be named whatever you want as long as its consistent.
If this is not working on your end, then something has a typo or is in the wrong folder, etc. Maybe provide some screenshots of your folder layouts and file structures?
Thank you for confirming 
I did it again and this time it does properly show the Chex Quest graphics. The first time I did this (exactly the same way) I swear it had Doom graphics and sound effects but with Chex Quest level layouts.
This is not ideal, but I guess it’s technically solved.