The hardest problem to overcome when trying to build a fire in wet weather is finding dry fuel. Even in the wettest weather, dry fuel can be found by splitting open a log and taking the wood from the center of it. You can get both your kindling and your fuel from this source. Another place to find dry fuel is on the underside of dead branches - especially those still on a tree.
Before you begin gathering your fuel, you will need to set up a dry place to store it as you collect it. A tarp can be used for this if you lay it out on the ground, place the wood on it, and fold the tarp over it. This will keep the wood off the wet ground and keep rain off it as well. If it is windy, you should place a few rocks on top to hold down the tarp.
Once you have your fuel, you can begin gathering tinder. If you have some with you, you're good to go, but remember, that because your kindling may be slightly wet, you will need more than the usual amount of tinder. If you need to collect it, there are still several options for finding some. Milkweed seed pods are fairly waterproof, and the fibers inside are pretty easy to ignite. Pocket lint is another possibility, but you may have difficulty getting enough of it to light your kindling. You can also make wood shavings from the same wood you're using as kindling. If available, birch bark can be lit even when wet.
Once you have your fuel, kindling, and tinder, you are ready to lay the fire. This is done as with any other fire. You may wish to place a tea candle in the tinder pile as well, as this will help keep things going long enough for the kindling to catch.
Once your fire is lit and the fuel is burning, you can lay wet logs next to the fire to dry them out before trying to use them. The heat from a good hot fire can drive the moisture out of even the wettest logs. You may need to turn them over periodically to dry all sides.
