I'm specifically want to know how people upload pet overlays, but I'm also wondering how users put those pictures on their forum posts.

Sorry is this is the wrong area. Just move it if that's the case.

If you're getting a custom overlay, you can upload it directly to the site. Otherwise you need to use an image hosting site like photobucket (which won't work for forum images), or tinypic or imgur. Personally I use imgur and I find it works pretty well.

first you have to upload the image to an image hosting site like imgur or tinypic. for pet overlays, unless you're doing a custom overlay (in which case you'll just upload the image directly to the site here), you'll use the following code on your pet's profile, and paste the image's URL in the indicated place:

{background-image:url(URL); background-repeat:no-repeat; height:200px; width:200px} if the image you're using is a different size, just change the height and width values to match.

for forum images, just paste the image URL on this page, but make sure the image does not exceed the stated size or else it will be cleared.

Do pet overlays fall under custom images?

A custom overlay is the one you have to pay to upload. A regular pet overlay that only shows on your profile would need to go on an image hosting site, and explained that part in better detail than I did :P