How to Make A Fire In The Rain or Wet Conditions
Gather some kindling, dead wood that is about as wide as your finger or less.
Pine will usually make a great fire starter, the sap/resin is flammable after it is heated.
Use a sharp knife/hatchet to strip away as much bark and wet wood as possible.
If you can use a hatchet to split larger pieces of wood into kindling, this will expose the drier inner layers.
Start a small fire using the stripped kindling.
Use the small fire to heat and dry the larger pieces. It will put off a lot of steam/smoke, but the smoke should subside after a few minutes.
It will take a bit more time and effort to get it going, but once you have a nice bed of coals established you should be able to keep the fire going even if it starts to rain again.