My system goes like this.
With the Flounder laying Brown side up, cut off the dorsal (top) fin from head to tail. Sissors work good for this. Cut just deep enough to take the fin off in one continuos peice. Now the cut you made should have seperated the Brown skin from the White skin. Do the same thing on the (bottom) fin.
Now, just behind the Gills, cut straight down thru the Brown skin, meat, back bone and "White side" meat, being careful not to cut thru the White skin.
Now turn the fish over, and with the head in one hand and the body in the other, pull the White skin all the way down to the tail.
Leave the White skin attached to the tail, and cut straight down thru the meat and back bone at the tail, being careful not to cut thru the Brown skin.
Turn the fish over and pull the Brown skin off from tail to where the head used to be.
Now the lateral (center) line is plainly visible on the skinned fish.
With the tip of a fillet knife, cut down thru the meat, following the lateral line, and being careful not to cut thru the back bone.
Starting in the cut you just made along the lateral line, cut the fillets off the carcase.
The key is finess, not force, and a sharp knife.
It's a lot easier than it sounds.