Using the mod:
Simply craft one and place it, then wield flint and steel and right-click on the campfire to ignite it.
If you want to cook with it, craft a grille and place it over the campfire/fireplace (whether it's burning or not). Add sticks and ignite the fire, if needed, then right-click on it with some food. After a short pause, the food is done, but get it off the grille quick, or it'll burn up into a puff of smoke!
A fire will burn down into embers after a while; just add more sticks by right-clicking with them to re-ignite the flames (it takes a couple of seconds for the flames to appear). You cannot cook over embers.
After a while, the embers will degrade into ashes -- the fire is completely extinguished now. Gather up the ashes, add sticks, and use flint and steel to re-ignite it.
Crafting and technical:
Mod dependencies: default, fire, basic_materials
Get it as a zip (VanessaE's fork): https://gitlab.com/VanessaE/new_campfir ... master.zip
...or browse the code: https://gitlab.com/VanessaE/new_campfire
Discussion / Issues: Discussion should take place here, on the forum post. Bugs should be filed against VanessaE's Gitlab fork (link above), or here in this forum topic if necessary.
Video (for old versions, but still useful):
On youtube: https://youtu.be/DEoYb-Jz0Mw