King and Emperor? My god I've no idea how you survive!
If you are near aggressive civ. leaders, have a small army (2-3 ranged, preferably seige, units) in the ancient/medieval area and focus on generating as much gold as possible, rather than expanding quickly. Keep a scout near the borders of the aggressive civs to see incoming armies. They will probably declare war sooner or later (stupidly thinking you are weak) so you wall up vulnerable cities and purchase melee units with the gold. Fortify them next to the cities, watch the AI send its armies to their doom and follow up by taking as many cities as you see fit (or happiness allows). At that point, you recovered the early resource advantage the AI's had, making the rest of the game easy.
If you are near peaceful civ. leaders, research seige weaponry asap, research swordmen, draft as many as your iron allows and take a peaceful civ's cities.
King and Emperor are easy, things get tricky on Immortal and Diety as the early advantage the AI has is huge, making some games almost impossible to win.
Is it just me though or does the new expansion tend to prevent Civ's from declaring war? I miss the days when the AI used to gang up on me and launch a surprise war which I'd have to battle my way out of. I used to play the hell out of the game but since the new expansion I've hardly touched it because of the lack of war. Not saying thats all I do, it just makes it more interesting than constantly buying City-states favour and dominating the UN.
Some leaders are very aggressive and they will declare war for sure if your army is considered weak (the Zulu one is notoriously psychotic). You must've played games in which you were placed near peaceful leaders (the Dutch, the Swedish etc). I have however noticed that "Great" wars, with multiple alliances/civs involved, happen less often.