Bicycle Mechanics - Caliper brakes not working.

I just replaced the brake cable (and housing) on a really old Centurion road bike but the back brakes just don't seem to want to work. Not only does it take half of the stride of the lever to get the pads to touch the rim, but there's not enough tension to keep me from pulling the lever all the way to the handlebars. Even then, the brakes don't work very well. When I do pull it all the way, it seems as though the (Dia Compe sidepull, by the way) brakes compress, it looks as though one of the calipers actually bends, is that possible? Or maybe the brake pads are just way too old to work (~15 years, I'm guessing) and they're compressing. But one way or another, something isn't working right and I just replaced the cable thinking that was the culprit. So what's the problem area at this point: the lever, the pads, or the calipers. That's all that's left! Thanks.

--
Will

I had the same issue on my bike which is also a centurion. Do you have the non-aero brake levers? I found a huge difference in braking when I switched to aero levers.
Couple of things to try (I did most of these in one day and it fixed any problems I had)
Clean rim (I used rubbing alcohol and a cloth)
Replace housing/cable (you took care of that)
Are you using any sort of brake tool when you attach the cable? I found it a lot easier to get the cable tight by using one of those and just yanking it through the guide in the caliper and tightening the bolt to hold the cable.
New brake pads can be had for 4$ if you are not particular about high quality stuff.
I did most of this stuff and then took my bike out and have found no problems with my brakes since.

I just started learning repairs when I got a beater to ride around town a couple of months ago so hopefully someone with more experience can help you more specifically but until morning this may help.

When installing the cable, try a couple of things. First, turn the barrel adjuster so that it's at least halfway through its adjustment range. Then get the calipers to the rim and hold them there. You might want to use a pair of large vice-grips to help you out... be careful though and maybe use pieces of old innertube between the jaws of the vice-grips and the calipers. Pull the cable through the clamp and tighten it down. When everything is released, the brakes should still be touching the rim. Let out some slack using the barrel adjuster so that the wheel spins freely without the pads touching.