I haven't heard anyone tell they do it exactly as I do....and I'm not saying my way is "right", either.
But....I make the cut inside the ligament on the back legs and insert the gambrel arms, there. From this point...I make my cuts around the skin at the hind legs where I want to save the meat.
I then use the gut knife to cut under the skin.....up the legs. The I peel the hams, downward. Cut around the anus.....then pull the skin down the deer's back. If you have to make small cuts along the way....no big deal....but I would NOT suggest using a filet knife. get something with a rounded tip for this exercise.
Make the same cuts on the deer's skin at the front legs (where you want to begin to save the meat)....and make your skin cut from the bottom on those, too (From the inside).At this point...you can likely pull the skin all the way down to nearly the deer's head.
A good tip is to leave the deer's head touching the floor while you're skinning/quartering. It'll keep it from "swaying" too much while you're working.
Now.....begin quarterin with the shoulders. It'll be REALLY easy. Then.....takeout your backstraps and inner tender loins. Now....you're ready to take the first ham. This one should be relatively simple. Your carcass is still stable. When you get to the last ham.....lower your carcass to the floor, some, and begin work on your final ham. KNOW that when you make your final cut on the last ham.....your carcass will want to fall. Just be ready for this.
Last ham is removed from the gambrel......saw the legs off at the point you made your incisionsto remove the skin.....and your deer is now quartered.
Put it on ice to take to the butcher.....OR....do it yourself.
Good luck