The first problem is, in your area, now is far from the best time to be
trying to establish a lawn, unless you are doing it from sod. Now you
will have need to use a lot of water and will have competition from
weeds. So, you may want to consider waiting till very late
summer/early fall which is optimal.

When you do it, first, I'd mow down what you have there with the mower
set as high as possible. Then I'd wait a few days for the weeds to pop
back up a bit through the debris thats left from cutting. Then use a
tank sprayer with Roundup or another glyphosate weed killer to spray
everything. I'd make the solution relatively strong, about 3%, so
you'll be sure to kill eveything. In about a week to 10 days you can
mow it again, this time as low as possible. Then rake up any debris.
Rent a slice seeder and use that to seed it. Apply a starter
fertilizer and keep it constantly wet. I'd also check the PH now and
adjust if needed.