From Civilization V Wiki
If you are the last player in possession of your own original capital you win. So if you capture all other civs' capitals and hang onto your own, you've achieved victory. However, this can be tricky. Suppose you're in a five-player game and you capture three of your opponents' original capitals, but the fifth player sneaks in and captures your capital while you're not paying attention – then he would win immediately. In other words, it doesn't matter who captures what: it's the last player holding onto his original capital who gets the victory.
If you've lost your original capital, but still possess other cities, you can still win another type of victory: culture, scientific, or diplomatic. However, you cannot win a conquest victory until and unless you recapture your own original capital.
Destroying an Original Capital
Simply: can't be done. A capital cannot be destroyed by any means. It can be captured, but not destroyed. Dropping a nuke on the city will at worst reduce its population to 1, but the city will not be destroyed. Accept it and move on. (Remember that you can drive another civ out of the game by destroying or capturing all of its cities, so you can still wipe your foes off the map even if you can't erase their capital from the face of the earth…)
Current Capital vs. Original Capital
If your original capital has been captured, another of your cities will automatically be assigned as a replacement capital. This city functions in all ways like the original, except that it can be destroyed, and it does not count towards a Conquest victory. If you ever retake your original capital, it will resume its leadership position in your civilization.
Whether this is accomplished by early offensive outings, or by build-up+late game steam-rolling, this is undoubtedly one of the most dangerous routes to victory.
The AI in this game seems a lot less capable of producing mass amounts of units for which they have no basis, and less capable of out-teching you at will.
However, this has been seemingly compensated by a innate survival instinct that causes them to forge an alliance if they see you trying to dominate via military power.
A good start to this Victory is to make a B-Line towards horseback riding and to quickly locate and settle your second city next to a horse resource (Hopefully in the 30 or so turns it will take you to build a settler, worker, and research the appropriate Tech, your starting warrior/scout will have located one).
Horsemen are fairly inexpensive, despite their strength advantage in early game fighting, and you can probably reach this technology and have quite a few of these units out and about before pikemen are in play by other nations. They are also gifted with a move after their attack, so you can either move them out of their latest victim's reach, or allow them to fortify and heal, or perhaps both.
If you are a fan of early aggression and this sounds viable to you, then your leader should definitely by Genghis Khan. His Mounted Movement bonus is epic for this approach, as it not only allows you to redeploy these units across the world at an amazing pace, but also allows you to have greater strategy advantage in their after-battle movement.
However, early aggression does not depend on beating them quickly. Keeping an opponent on the defensive is also a viable way for you to reach a much more plausible victory point of out-teching your opponents. After all, 3 Infantry will triumph over 8 Longswordsmen with relative ease. Reaching these critical military technology points is crucial to achieving sure victory.