What to add depends on what the soil needs. How much organic matter is in that soil now? Perhaps these simple soil tests will be of some help,
1) Soil test for organic matter. From that soil sample put enough of the rest to make a 4 inch level in a clear 1 quart jar, with a tight fitting lid. Fill that jar with water and replace the lid, tightly. Shake the jar vigorously and then let it stand for 24 hours. Your soil will settle out according to soil particle size and weight. For example, a good loam will have about 1-3/4 inch (about 45%) of sand on the bottom. about 1 inch (about 25%) of silt next, about 1 inch (25%) of clay above that, and about 1/4 inch (about 5%) of organic matter on the top.

2) Drainage. Dig a hole 1 foot square and 1 foot deep and fill that with water. After that water drains away refill the hole with more water and time how long it takes that to drain away. Anything less than 2 hours and your soil drainsâ too quickly and needs more organic matter to slow that drainage down. Anything over 6 hours and the soil drains too slowly and needs lots of organic matter to speed it up.

3) Tilth. Take a handful of your slightly damp soil and squeeze it tightly. When the pressure is released the soil should hold together in that clump, but when poked with a finger that clump should fall apart.

4) Smell. What does your soil smell like? A pleasant, rich earthy odor? Putrid, offensive, repugnant odor? The more organic matter in your soil the more active the soil bacteria will be and the nicer your soil will smell.

5) Life. How many earthworms per shovel full were there? 5 or more indicates a pretty healthy soil. Fewer than 5, according to the Natural Resources Conservation Service, indicates a soil that is not healthy.

Here's my soil, like I said, clay. I'm going to begin overseeding during the winter so maybe I will go ahead rototill some good compost and manure come fall. What should I do for fertilizing this summer?

I don't see any organic matter in that test sample and that says grass will have a hard time growing for many reasons, lack of available nutrients, lack of moisture, etc.
Which, if any, fertilizer to use depends on what your soil needs.

That test looks like 99% sand and 1% silt with 0.000% clay. If you had any clay at all you would not be able to see through the water at the top.

Although this might surprise you, this is extremely common finding from the jar test. Most people do not have clay. If you send a sample to Logan Labs in Ohio they will tell you how much magnesium you have. Too much Mag can give even pure sand the effect of clay. But if you treated your sand as if it were clay, you could mess things up further. For example you might use gypsum instead of calcium when you have too low a pH. Oh wait! You already added alkaline gypsum to your alkaline soil...which may already have too much magnesium in it.

Other bermuda lawns in Phoenix are already greening up. I'm not suggesting you are behind, but it could happen any day now. Go to the Organic Gardening forum, find the FAQs, scroll to the bottom of the list, and read the Organic Lawn Care FAQ. That will give you some ideas about how modern organic lawn care is being done. Then come back and ask questions.

The hydrolyzed fish is a waste of money. You would need about a gallon per 1,000 square feet to get any serious benefit. Worm castings are just expensive compost. Another waste of money. There are a lot of products in the organic arsenal. Only a few have any value.