I recently finished a space race game with Ramses on Emperor (save attached). This was my second attempt at Civ5 and the first game in Civ5 that I have finished. It was a piece of cake, really, and I had no idea what I was doing, basically fumbling through my second time in the game. I made all kinds of mistakes and finished late, and I was a little disappointed that I could beat the game on Emperor on only my second try. On the other hand, the game does look beautiful, and the aesthetic experience of the game is terrific. But I digress.
Anyway, a few observations that may be relevant to this King level game:
1) I was able to build all the wonders I wanted without marble (till late) or the 33% policy bonus (I pursued Liberty first). You can build any wonders you want with little difficulty on King. You can play a chill game and still do this. Not a problem.
2) Your only concern will be an AI invasion. They won't launch before you, even if you play a flawed fame. At least they didn't come remotely close in the game I played, and there were some peacemonger-type civs. A big question in the game is the right balance between size of military and cost effectiveness for the empire.
3) Keep you empire small at first so that you can build the science-friendly national wonders early. You need a library in each city early.
4) I recommend Liberty, then Commerce, then Rationalism for policy trees (though I am still a total beginner obviously and may very well revise this opinion). No need to fill out all the policies. Just take the ones that further your science the most quickly.
5) I recommend early Civil Service for the food bonus.
6) Don't attack any City States. The AI will hate you for a long time. The diplo penalty for attacking even a single city state is not something to be trifled with if you want a fast finish. I trifled, and it cost me more heartache than it was worth.
Hope this helps. Not sure if I will play this GOTM, since I just finished a very similar game on a harder difficulty level. I'm currently trying an Immortal-level game as the Iroqois, and it is much more challenging and interesting than the last one.