Take a few cotton balls and work some Vasaline into them with your fingers. Put these in a sandwich bag. Use one per fire.
Find a big rock, or clear a small area of ground of leaves and debris. Place some rocks around this to create a campfire ring. In the middle of this, put some leaves etc so youre not starting your fire on cold, damp ground. Make a small teepee out of very small twigs. Pull one of your cotton balls appart a bit and place under the teepee. Light this. Once the twigs are burning well, put on a few larger twigs, want until these are burning well and repeat. Don't go too fast, as the cold twigs will suck the heat out of your fire until they get started.
Don't burn big, huge logs. 3 or 4 inches in diameter is about as large as you want to go. Make sure your wood is dry. Use deadfalls, do not cut down live trees or used live branches off of trees.
After your fire is out, pour water on the area, scatter the rocks and sweep some leaves and forest debris over the area of your fire pit so nobody knows you were ever there.