Did it starting out on Easy and working my way through, skipping straight to the boss conclusions on the appropriate chapters. I wasn't sure whether I would have to repeat chapter 3 or not, since though I got under 10 deaths, it didn't award me any grade at all. Weird. Anyway, I got as far as the Big Giant boss and after I got my A, the achievement popped without me even having to do the Little Tailor battle
! I can only surmise that this was due to my already having got an A for this level previously and the game must have remembered.
So as mine and rdowdy95's experiences seem to show, you don't have to do the whole lot through in one lick, but can maybe redo the ones that you are certain that you haven't got top marks for?
Anyway...with that done it will be trying to do the poxy chest achievement. I'll take a deep breath and give it a go...maybe next weekend