I use Dawn. I use a washing machine, cold water, gentle cycle. I only wash coyotes in this manner.
That grease at the bottom is the accumulation of oils in the rest of the hide that you pushed to the bottom, or just settled there as the coon started drying (my theory, but can't think of a better explaination).
Try this just "gags and giggles". Put your hide on a board (coon hide) after you fleshed it. Take a "straight edge" of some type and scrape it down the "finished product" and see if a table spoon or better worth of grease in your "finished product" doesn't come out. For this I have an old draw knife with a straight blade, but the hoe shaped fleshing tool gets the trick done.
By doing this you will be amazed at how much grease you leave in the coon when you flesh it.
Now, the final step I use in finishing out my coons: Once you determine that the coon is dry enough to take it off of your stretcher (be it wood or wire) stack them between sheets of cardboard. Put a sheet plywood on top, and something reasonable heavy on top of that to "compress" it all. In about a week you can remove them, fluff your windows with a little laquer thinner and put them on hangers. Look at your cardboard. Note the grease, now shake your coon, notice the rattle, now look at your fur check, notice the extra dollars. Sound time consuming? Nope, not at all, an extra 30 seconds after fleshing, an extra 30 seconds after removal from the stretcher.
You don't want to do this on the floor, use a table.