Navigate to your mapname folder and into your zone folder - "Call of Duty Black Ops III\usermaps\zm_mapname\zone" You'll find both "loadingimage.png" and "previewimage.png"
Loading Screens and Preview Images are set up slightly different. Loading Screens need to be replaced every time you make a new map, however Preview Images can be set up individually. We'll go over Preview Images first.
Open your preview image into whatever photo editing software you use and edit it to your hearts desire. Don't change the name. Save and that's it!
Open your loading screen image into whatever photo editing software you use and edit it to your hearts desire and save it. Don't change the name. This is where it's different. Open APE and find the image "black". Not "$black", just "black". Replace this image with the one you just created inside of "Call of Duty Black Ops III\usermaps\zm_mapname\zone\loadingimage.png". You're going to have to do this every time you make a new map.
Make sure your image settings in APE are set up exactly like mine (besides the actual image):
Next, find "core_gfx.csv" inside of "Call of Duty Black Ops III/zone_source/all/assetlist/core_gfx.csv" and find this within:
Now comment it out so it looks like this:
Lastly, open your mapname.zone file and insert the following:
//Preview Image image,black
Save, and that's it!