Welcome, rx.
Looking good. Desert King can only produce a breba (first crop) unless you were in Southern California, but it's a productive and hardy variety for your climate.
Branches that are green right now (this year's growth) will give you fruit next year, so when pruning, take that into consideration.
Any dieback should be cleaned out. Those dead tips will stimulate more branching, which will give you more fruit.
Figs like slightly alkaline soil, so a little shake of pelletized lime (the stuff you use for lawns) two or three times during the growing season will allow your plant to efficiently absorb the nutrients it needs.
Osmocote, Miracle Grow, Espoma or any well balanced fertilizer with micronutrients will make that plant happy, though it looks like it's pretty happy already.
If you plan on bringing it inside for the winter, you can fertilize it as any house plant. If it's going to spend the winter outside, you should stop fertilizing by the middle of August, so the new (green) growth has enough time to harden off.
Any stems that are bright green are really sensitive to cold temperatures, so you want that plant to slow down or stop its growth by the time the leaves start turning and falling off.
Try doing a search for "pinching", a practice that forces branching.
Again, welcome to the forum.
Good bunch of people here, a kind community.