Is there a good way to put biochar into a forest, or should it be in there before you plant the forest?
Be gentle with an existing forest.
If your trees are already in place, then you may put the biochar directly in place around the trunks of the trees you want to help grow. You may rake the soil to put the biochar beneath the normal humus forest ground, to avoid it to be washed away to quickly.
Depending on how you created your biochar (it may contain more or less ashes, that can change soil Ph a lot), you should go smoothly with it. Start gently on chosen spot for 2-3 years, then see how it goes for those chosen places, then extend to other places step by step.
Don't do it in place:
One method of making biochar: pile up woody debris in a shallow pit in a garden bed; burn the brush until the smoke thins; damp down the fire with a one-inch soil covering; let the brush smolder until it is charred; put the fire out. The leftover charcoal will improve soil by improving nutrient availability and retention.
This probably will burn up tree's roots and arm them, if it doesn't kill them (let alone the fire issue for the whole forest).
Biochar should not be added to any forest soil or natural setting. The soil that evolved there has the fixed amount of carbon it requires. Biochar is for repairing damaged soils or adapting agricultural soils. Despite numerous sayings of "Biochar" in the amazon rainforest, that was not biochar, it was "Terra Preta" which was meant for agricultural soils and subsequently after their extirpation the forest simply grew back.