What's the best way to install casing around a door frame that is in a double drywalled wall? Since the frame is not as thick as the wall, isn't there an issue with attaching the casing when the wall and door frame aren't in the same plane?

You just have to pack out the jambs that's all. Run a level or board along the wall and let it hang over where the jamb is. Measure from the jamb to the level/board. Take a measurement at the floor, in the middle, and most importantly at the top. It should be pretty close all the way up unless you have a seriously screwed up wall :) Then use a table saw or power saw and rip the wood to the width you need.

It'd pretty consistant from top to bottom. The outer layer is 5/8", so I was thinking of maybe just attaching a piece of 5/8" filler around the primeter and then adding my casing.

Good. Are you staining or painting the frame? If you're painting, your butt joint doesn't have to be as pretty, but if you're staining you want the joint to be as tight as possible so you don't have to fill it. After attaching the extensions, if you find any spots where the jamb is sticking out you can always just use a block plane to level it out. On the other hand if you find any spots where the drywall is sticking out over the jamb then you can always tap the drywall in with a hammer. You have the width of the trim to play with.

I'll be staining the frame.

If you didn't order a door with a wider jamb to match the thickness of the wall you can build a jamb extention to get it to full depth. Basically just a 3 sided box that butts up against the jamb bringing it to full width. You just want to leave a 1/8" reviel to make it look good.