Look for dead standing trees. When they stand up they dry. Rain and snow usually only wet the outside. Cut a log and split it. You will find the driest parts in the middle. You can shave off some with a knife to start the fire with. Then you can split the rest into 10 to 15 millimeter thick pieces to get the fire going. If there is birch bark around, use it. It contains a lot of oil and therefore is easier to light than many other things when it's wet.
Once you have a fire you can lay thin branches around it to dry.
Take a look at this video for a demonstration of wet weather fire starting.