Line Your Backpack
Line your backpack with a heavy-duty white trash bag (45-gallon size), and fit your pack with an external pack cover. Use a white trash bag because it is much easier to see your belongings inside than if you line it with a black trash bag. This is recommended by both "Allen and Mike's Really Cool Backpackin' Book" and "Mountaineers: Freedom of the Hills 7th Ed." Line your pack even if it is made out of waterproof material. Some pack companies such as Osprey design their backpacks to come with an external pack cover, which fits over the pack with elastic and is made of a waterproof material such as siliconized cordura. These can also be purchased in a range of sizes from day pack to expedition-size packs. Sea to Summit, Outdoor Research and Gregory make fine pack covers.
Equip yourself with good shell layers that are waterproof and breathable. Brands that are well known for making the best rain shells are Arc'teryx and Patagonia, and are worn by many backcountry guides from the rainy region of southeast Alaska. Anything made with GORE-TEX XCR or GORE-TEX Pro Shell is fantastic. Make sure your rain jacket has long-enough sleeves, a big enough hood and sealed zippers. Make sure your rain pants can be put on easily over shorts or other pants in a hurry. Full or partial side zips are preferable. Arc'Teryx, Patagonia, Marmot and Mountain Hardwear all make superb rain pants.
Protect Your Goods
Place your sleeping socks, extra base layers and sleeping bag (especially if it is goose down as this does not retain its ability to keep you warm when it is wet) in yet another bag inside your lined pack. A simple trash bag works for these. Consider a synthetic sleeping bag if you know you are going to be somewhere it rains a lot; these retain their insulating capabilities regardless. Also, consider keeping your first-aid kit in a plastic Ziploc bag. These techniques are used by professionals in many outdoor programs such as the Adventure Education program at Plymouth State University and the Outdoor Studies Program at the University of Alaska Southeast.
Protect your navigation materials such as the map you are using with a clear plastic Ziploc so that you can have it out in front of you when need be. If you want to spend a little extra you can have your map laminated, but after a while the corners can peel up and the seams can crack allowing water inside. It is much easier to fold it to the quadrant you are using that day and place it in a Ziploc. Pelican makes dry boxes in many sizes, from tiny ones that will fit just your cell phone and wallet to ones that will fit a large SLR camera. These are often used by kayakers. If you know you will be making several river crossings you may want to put your most valued goods such as GPS, cell phone, wallet or firestarters in one of these in case the pack goes down in a river.
Keep Your Feet Dry
Wear hiking boots that are made with GORE-TEX or other waterproof materials. If they are old and cracked around the bottom, seal those cracks up with seam seal that is used for sealing tent seams, and while you're at it, seal those tent seams up. Wearing waterproof gaiters can also help to keep your boots and pants dryer. These also help when traveling through mud country.
Renew Your Waterproof Materials
After repeated use in the weather, waterproof materials such as your rain shell and pants can become less repellent to rain. To keep your gear as rain repellent as possible, use a concentrated treatment for waterproofing synthetic fabrics. Brands such as Nikwax or Granger's both make spray-on and wash-in formulas to help rejuvenate the repellency of fabrics like GORE-TEX or eVent. They also help to repel stains. The shoulders of rain shells are usually the first to need treatment due to the pack straps rubbing on the area. There are also spray-on boot treatments to help keep your boots as repellent as possible, such as Grangers G-wax Fabric and Leather Waterproofing.