Probably still hard to beat Khenarthi's sheep--but if you just play though Thieves guild and steal everything you see, you'll hit your fence/launder limits easily.
Pickpocket & Steal - Then fence items. Unlock Legerdmain passives and put skills point in them as you level up. Where gear that enhance sneak and use invisible poisons when necessary. Avoid getting caught if you can. It will become easier as you level up Thieves guild and Legerdmain.
Pickpocketing achievement needs to you pickpocket 1,000 people so might as well work on that at the same time. Then use Blade of Woe and find a new target.
You can go the the docks in Daggerfall and steal everything inside the ships. No witnesses inside the ships but if you are stealing while on deck sometimes a NPC on the docks will see you.
You can fence the items near by so it doesn't take much time to max out what you can launder/fence each day.
and then the parrot said, "must be the water mines green too."
I believe it was @BlackSparrow that suggested the two ships in Daggerfall in another thread (which I can't find right now), which works just like a charme. It's so easy it's criminal. I like to pay the castle a visit as well. I'm courteous like that. Equal opportunity and all.
Real stupidity beats artificial intelligence every time. ~ Terry Pratchett
There is a post about how to level Ledgermain already, by stealing apples in Craglorn. Fence AND Launder your max capacity each day. The post, which you can find by doing a search on the forums, explains how the point system works for leveling Ledgermain.
So, go back there. Orsinium is a thieves wet dream.
The Ships in Daggerfall are good if you grab the provisioning ingredients (laundering levels Legerdemain too) but you can fill your bags in Orsinium too.
Enjoy the game, life is what you really want to be worried about.
PakKat "Everything was going well, until I died"
Gary Gravestink "I am glad you died, I needed the help"
The fastest way is to steal everything, including the small worthless stuff like food in pots, bags, and containers. Even stolen food gives you points which add up. Ive got characters with 10+ ledgermain just because they sold stolen items.
Put points into the skill that increases pickpocket chance as soon as its available as well and the amount you can launder per day.
Find a place with very few or no guards. Steal from all containers Pickpocket everyone in sight. Meet your daily max of Fenced and Laundered items. Every day. I repeat: Every Day.
AFAIK all items give the same XP, regardless of price to launder/sell, so as people said, steal anything and everything, plus invest in the perk that allows more transactions p/d.
Picking locks, doors, chests and safe boxes
Pickpocket, everyone
Stealing, everything not nailed down.
Make run in wayrest, shrine to windhelm and make a run, shrine to Skyreach in auridon and make a run, shrine to mournhold you get the picture. Hews Bane also has alot of shops and houses.