The thieves guild is a good way to do that, or just do some radiant quests. I have 110000 septims in Skyrim from delving into dungeons while doing Azra's Staffs for Neloth (which also gave me money), although I'm level 55 and I've played 200+ hours on that playthrough.

Edit: I forgot about the non violent part, but the Thieves Guild doesn't abide violence anyway, and if you're really good at Illusion or Sneak, you can make dungeon diving fight free.

Exploits.

In Oblivion, you can make a lot of money with alchemy. Harvest all the ingredients at Odiil Farm, Shetcombe Farm, Lord Drad's Estate and the Skingrad Vineyards, and turn them all into restore stamina potions you can sell.

Depending on how good you are at mental gymnastics the Dark Brotherhood questline works too. You are not "killing" people, you are just "helping them find peace."

^Find peace in the Void?

@Pearl Lunar Hail Sithis