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I recently did my own differential fluid service and decided to make a video for anyone that wants to do it themselves! I used the recommended fluid highlighted in the latest GM TSB for differential services.
The fluid you need is Dexron LS Gear Oil 75w-90 #88862624, you'll need 2.28 quarts for a base C6. I was able to get it on amazon prime for ~$15 a quart.
The procedure should be the same for a Z06, the only difference I know of is that the Z06 has a small retainer clip on the plugs.
It's very easy and only took me about an hour and a half, that was including filming and waiting for the old fluid to run out of the diff!
Hopefully this helps anyone wanting to save a bit of money and/or wants the satisfaction of doing it themselves!