How to play against a friend with higher demographics?

I'm starting to believe Civ 5 really makes this hard, as once you get behind in tech/GNP you will lose the ability to beat them.

The situation is this, I've started on a not too prosperous continent (other civs border-to-border) while my friend, the opponent, has started up on a much more free continent. He leads the demographics but not in an order of magnitude. I run at third on every demographic attribute. Are there any tips and tricks to become victorious in an unbalanced situation against a human player?

You've got just a couple of options, generally.

One is war. War, war, war. Crush your neighbors. Hopefully you're not so far behind the AI that you can't overtake them. The AIs are terrible at war (as I often mention) so just put siege units in your border cities with front-line troops between them and the enemy. Declare war, let them throw themselves on your swords or what have you, and when they beg for mercy, take their stuff. Lather, rinse, repeat.

I tend to puppet rather than annex, as that reduces the happiness hit. You can also raze and build on their turf, although this works best for tiny cities that are well placed. You'll also have to work fast if the AI could potentially settle in the cleared area before you get the chance.

The other possibility is to go for a sneaky cultural win. Cultural policies get harder to get the bigger you are. However, this can be difficult to do, depending on how far behind you are and how small you are. It's also harder in Civ5 than it was in Civ4, since your culture doesn't steal hexes in Civ5. (Do note: If you're playing with BNW, you need tourism, which is a totally different subject...)

Most of the other options are stymied by the fact that either your size limits you, or the AI will start to work against you as you approach victory.

You have to press any technological or territorial advantage you can get. If your continent is rich in iron and your friend's continent is not, you can dominate the seas with frigates and try taking some coastal cities to contain him through the Renaissance and Industrial eras.

If you don't have iron, build up Privateers and coastal ranged units to repeal naval invasions. If you are on BNW, you could try going for Ironclads. These will crush any naval unit of the era, but will cost you coal which is relatively scarce. You could also rush to submarines. With enough submarines, you can keep him far away from your continent.

If you're really lucky and you have a lot more oil than your friend, then Battleships and Bombers will let you take the win.