Still find it difficult to keep up with the AI (Continents) in the late game. If you can continue to steamroll your opponents militarily, e.g. on a Pangea or on two continents separated by shallow waters, you can build an empire bigger than the AI. Otherwise, one AI swallows up each continent, and with its happiness bonus it can have a higher population, supporting more science, more units because of less upkeep, and it really just feels like a long unpleasant slog 'uphill'.
I also resent the fact that the only sure way to win at King or above is to conquer everything in sight and raze the majority of cities ... I've tried small-civ culture wins and they just aren't feasible, you end up strapped for cash and the AI builds more culture than you with its dozens of cities anyway. The national wonders are not adequate compensation for having a small civ at higher difficulties.
Anyone suggest any way out of this mess? At the moment it's looking to me like there's only really one way to play Civ5 at the 'proper' difficulty levels, and that has diminished my interest in it greatly. Same thing happened with Civ4, although it had other options like religion that reduced the monotony of constant warfare.
I very rarely attack anyone, and even when attacked I generally play my wars defensively, only once in a blue moon actually capturing a city.
Yet I win most my games at Emperor (have also won on Immortal and Diety with similar methods, but prefer Emperor), playing most frequently on Continents.
I tend to prioritize happiness, production and grabbing good city sites as early as possible (in my current game it's around 1200 AD and I have 13 cities -- standard size map). I find science and cash will follow if you have plenty of cities, with plenty of pop and high production (you can quickly build all the appropriate science and gold buildings). Also, I space my cities fairly far apart (often 5-6 hexes between), so I wouldn't consider it ICS as I'm trying to claim nearby good city sites (luxuries, resources, natural wonders, rivers, etc). I often put an emphasis on culture (even if I mean to win by other means, I like to get plenty of social policies).
I feel that most people underrate the ability to play this game as a builder.
I don't have definite advice on build orders, as I tend to play it by feel rather than bothering with any grand strategies, but I can attest that it is by no means necessary to focus on military to win.
In most games, I'm also able to remain friendly with the majority (and sometimes all) civs -- I haven't tended to have the issues others have with diplomacy. I play this by feel as well. I still try to have in the ballpark of at least one military unit per city (more, if I have fewer cities) -- if you completely neglect your military than it's hard to complain if others decide to take advantage.
Rather than giving up, I would try various building strategies and see what works best for you. Perhaps practice these techniques more on Prince before attempting to move up.