First things first, find your Gmod folder.

This is usually at either
C:\Program Files\Steam\steamapps\common\GarrysMod\garrysmod
for 32 bit systems, or
C:\Program Files (x86)\Steam\steamapps\common\GarrysMod\garrysmod
for 64 bit systems.

If you're not sure, try the latter. If that doesn't work, try the first path. (It might be neither of them, but only if you manually changed your Steam installation folder, which I assume you only did if you knew what you were doing).

Now it gets slightly more complicated!

After you've found the correct folder, delete the 'Backgrounds' folder in there. Trust me, nothing can go (irreversibly) wrong here. Make a backup of it if I still haven't convinced you. Now that you've deleted 'Backgrounds', open the 'screenshots' folder. Copy the file path (at the top of the window, the 'address bar'), this will make the next step easier.



Now for the technical part! Open up your start menu, and type 'cmd' in the search bar. (If you're on Windows 10, the search bar is opened by clicking the magnifying glass on your task bar instead). This should allow you to open CMD.exe. Right-click CMD.exe, and select 'run as administrator'. CMD should ask your permission to run and modify files. Allow it to do so.

Now that the command prompt is open, type in 'mklink', followed by what you copied to your clipboard earlier. Be sure to put the copy-pasted part in between quotation marks! Then, replace the word 'screenshots' with 'backgrounds'. Press spacebar, then paste what you copied earlier again, and put it between quotation marks again. Now, the entire thing should look something like: mklink "C:\Program Files (x86)\Steam\steamapps\common\backgrounds" "C:\Program Files (x86)\Steam\steamapps\common\screenshots".



Almost there, just press enter. Now, there should be a new folder in your gmod folder, called backgrounds, If everything went right, this folder should have a folder with a shortcut arrow in the corner as icon. If that's the case, you're done!

Yes, I know I have a lot of screenshots...

What we just did, was make a so-called 'NTFS symbolic link'. This acts as a regular folder, using the name of the shortcut, while redirecting to another folder. We're beasically tricking gmod into believing our shortcut is the backgrounds folder, which Gmod uses to pick main menu backgrounds from. Instead, whenever any program tries to open our shortcut, it redirects them to the contents of 'screenshots'. This means that whenever Gmod tries to acces the backgrounds folder, it instead reads the contents of the screenshots folder, which it then uses as a background. If this sounds too technical, then don't worry, the important thing is that it works, and I'm sure that a quick google search or two will clear things up for you.

Now all you need to do is make some nice screenshots!

Well, they'll be lost in the process. This is just temporary, though! You can bring them back into the mix in three basic ways:

1. Wait for an update
   Garry's mod updates fairly frequently. Whenever there's a new update, all the default backgrounds lost in this process will be put back into the sycle automatically. You'll just have to wait a bit.
   
2. Don't wait for an update, force an 'update'
   You might know about the dev version you can opt into. I'm not going to explain it in great detail here, but you can look it up if you're that impatient. If you opt in, launch the game, then opt out again, you'll also redownload the backgrounds.
   
3. Copy-pasting them.
   Should speak for itself. If you decided to make a backup of the original backgrounds folder earlier, then you can now simply paste the contents back into your screenshots folder (or 'backgrounds' 'folder', same thing now anyway).