A Remington action will be a lot lighter than a Winchester and will be easier to get the weight you are looking for. That said I prefer the Winchester action and you can still get pretty light using one. Just stay away from Weatherby or Howa if you want light. Savage actions are pretty light as well.
Here are some weights to look at
http://brownprecision.com/SelectingComponents.htm
If you want light, forget about B&C. Their Medalist stocks weigh between 36-40 oz. The Medalist stock is a good one if you want a heavy rifle, some of the non-Medalist stocks are a little lighter, but are junk. A McMillan Edge is about 2X more expensive, but is the easiest and most cost effective way to get a truly light stock, 20-23 oz depending on style. You can buy stocks as light as 15-16 oz from companies such as MPI, but cost is much more than even McMillan.
www.hightech-specialties.com also makes a decent stock at around 24oz that actually sells for just a little more than the B&C, but you have to do the final fitting, add a recoil pad and paint it yourself. I've done 1, it ain't rocket science, but you don't save that much after buying a pad, paint and your time over a McMillan.
You should be able to find a Remington Mt rifle or Winchester Featherweight used for around $400-$500. A McMillan Edge will cost you $518. (if you're serious, PM me and I'll tell you how) You can sell the factory stock on E-bay for $100 or so and be under your budget and have a rifle that weighs around 6 lbs, maybe a little less before mounting optics.
Or you could add $100 to your budget and just buy a 5-6 lb. Kimber from the factory.