Here are some merchants you can sell stuff to:
Abernathy Farm - Connie
Drumline Diner - Trudy
Diamond City - Major hub/Many vendors
Good Neighbor - Minor hub/a few vendors
Grey Garden - Supervisor Green
Bunker Hill - Minor hub/a few vendors
Also vendors will visit your settlements (such as sanctuary) on occasion.
You will get a marker for Diamond City relatively early on during the quest:
When Freedom Calls
I'm more than a day into a playthrough with a stealth ninja melee character. I get these animations fairly often.
I'm primarily using General Chao's Revenge and Blitz to melee guys from far away, sometimes behind, sometimes not. Sometimes I sort of chop their heads off, sometimes I impale them through the chest, sometimes I jab the sword straight downward through the top of their ribcage near the collarbone. That last animation takes forever.
I can confirm that they happen in VATS when stealth attacking humans.
I believe I've also got such an animation at least once out of VATS, but I am using VATS almost exclusively. I recall getting these animations regardless of the angle at which I'm facing the enemy, but note that VATS places me behind them when it executes the attack. I don't recall ever getting such an animation on a non-humanoid enemy.
I find these animations annoying, and wish I could turn them off. They delay being able to act again for a while after leaving VATS, which has gotten me killed more than once when there was another person around I didn't see before attacking.
Assuming that you've met the criteria for a backstab animation to be possible, the animation actually occurring seems random. Sometimes it happens, sometimes it doesn't.
It's possible the animations may only trigger with certain weapons. Try a combat knife or a Chinese officer's sword. I've gotten them with both of those for sure.
Best Answer
Obviously there are ways to increase your income, but I'm considering that outside the scope of this answer, and focusing on getting better prices.
Discounts
Some vendors will give you discounts for actions taken in the game, like Carla giving you a discount for making a sarcastic remark when you meet her. Shopping at vendors that give you a discount will help.
Raising Charisma
Much like raising intelligence for additional xp, you can raise your charisma for better prices. I keep a set of charisma gear on me, including clothes (+1-3 CHA), a hat (+1 CHA), and glasses (+1 CHA). I put these on before speaking with vendors, and it improves both buying and selling prices.
As an experiment, my character with 3 CHA sold some items without the CHA equipment (3 CHA), then I loaded and sold the same items with the equipment (7 CHA). I got approximately 200 caps without, and approximately 250 caps with.
You could also drink some alcohol first, which should pay for itself.
Perks
There's a perk for better buying and selling prices, called Cap Collector. I haven't tried this, but the exact numbers are listed on the fallout wikia.
You can also improve buying and selling prices with the perks you receive from finding Tales of Junktown Jerky Vendor magazines.
Other Items
Some specific items will give you a boost specifically to prices. Grape Mentats will give you 10% better prices and +5 Charisma for 8 minutes.