I have 7 out of my 10 original villager pictures
i find them very easy to get, and i honestly believe the only quick method is to talk to them everyday, and then after doing about 3 favours for them, reset for their pic
if you don't reset for their pic, the pic will always be just one of many items they could possibly give you as a reward, so its unlikely you will get it
Only send one letter to them at all, thats all ive ever done, and i have 7 pictures..
Once they start sending you letters randomnly, you have a shot
When they ask for a fruit, give them a perfect foreign fruit
When they ask for a fish, give them a rare fish such as a napoleon fish, or a shark etc
When they ask for a bug, give them a golden stag
Before giving them the reward, save and continue
give the item over, if its not their picture, restart the game
continue until they give you their pic.