So, I've played quite a few games on prince difficulty so far, and have watched a craptonne of videos (I'm more of a visual learner). Bottom point is that I more-or-less know what I'm doing. Out of the prince games I have played, I won 3 and stopped playing about 5 because they were tedious wars and were a sure win. But in all my games, the things that were driving me crazy was the lack of AI money for trading resources and making research agreements possible; at prince diff., it's more or less impossible. People are saying King Is significantly harder but when I saw the in-depth changes between diff. Levels, it didn't look like much.
TL;DR: When did you move from prince to king (not a date, but how many games you put into it) ?
I moved from prince to king after about 5 win on prince.
What was your personal reason for moving out of prince (other then the obvious

)?
Just the obvious reason. Nothing special.
How did your first game go on king?
Terrible.
I was the Egyptians and the the Aztecs, Danish, and Greeks constantly attacked me to plunder my sarcophaguses and wonders. I could be progress in any way, so I evenually quit around the Renascence.
What to expect from king difficulty?
Honestly, if 10 games + watching LP series isn't enough to move on, how should I improve?
The think with Prince is that that are may strategies/playstyles that would give you great success in Prince, but will fail miserably in any higher difficulty.
For example, may playstyle is prince was very passive and very focused on building:
- High priority on Wonders: I try to build as may wonders as I can
- Lean military: very few units in peacetime, but will spend to upgrade to most recent tech.
- Peaceful: I only declare war if a Civ has a resource I disparately need. I try to play nice with the other Civs through diplomacy and trade as best I can.
- Build, Build, build: I usually build a buildings/wonders instead of unit.
Also, I was pretty aimless with tech: When it can to choosing a tech, I'd just choose the one that would take the least time to get.
This playstyle works on Prince: in the beginning, I'd be harassed by warmongers due to my small army and high number of wonders, but after by tech skyrockets above them, even my small number of units will have a higher military rating than their forces and they would leave me alone to do as I please.
On King, this strategy is garbage. Again, it'll get you in constant agonizing wars, and even if you get past that, you are all but guaranteed to have a runaway Civ take the game (usually with Science, if not, diplomacy).
To summarize, it all depends on how your strategy in Prince translates to the higher difficulty levels. I'd say that if you watch vids, then you are ready. That what I did when I got tired of getting whupped in King (after getting great advice from guys on this very forum) and to me, King is now easy.