This question might be framed backwards: all cities were capital cities until polities got big enough and complex enough to support multiple cities. Maybe it's more helpful to ask when the concept of non-capital cities started. When did the power of a city first begin to extend beyond its immediate surroundings and over other cities?

edit: typo

There are very few cities that were made to 'be the capital' and even fewer built to be the capital. From my studies, Ancient China has had a few 'capital cities' and all are based upon where the king (or later, emperor) resides/governs in. The word 'Capital' (首都) in Chinese simply meant 'First City' or 'Head City' (首 means the first or the head of and 都 simply means a very big city/metropolis) and has a pretty strong connection with the head of the state or the emperor.

Differ from the other nations, China's Emperors have a weird habit of moving around their capitals and does not usually choose the largest cities with the best infrastructures as their administrative centers.

I believe this is the same more or less world wide, 'Capitals' came to being as a natural progression of the increasing reach of the kings and warlords of the area. Once their reach goes beyond city states, the city that the central government resides in automatically became the 'capital' or whatever fledgling nation state it belonged to.

http://en.wikipedia.org/wiki/Historical_capitals_of_China

However, I don't think its until the colonization of the new world that a city was established 'greenfield' for the express purpose of being the capital. Quite a few national capitals are built this way in the Americas (Washington, Kingston, Brasilia etc.). With the exception of Japan. Then again, Japan is weird as hell.

http://en.wikipedia.org/wiki/List_of_purpose-built_national_capitals