My experiences with Bermuda grasses in Texas and California, is that it will die back to looking dead in winter and will come back with a vengeance in summer or when the soil heats up... It needs fertilizer and water to do it's best...
I personally don't like it because it is very invasive, has these harder than heck roots that go way, way, down, and are pretty much impossible to pull out... And then you have the seeds that fall off and are ready to go next season..Bermuda grows underground horizontally and pops up easily everywhere... It can get really dark green if fed a good fertilizer a couple times a year and watered in, and of course, watered regularly..
It has been my experiences that weeds Will grow where nothing will. And they will try to grow in the driest, most neglected soil... Because that is their job.. Some weeds actually produce nice looking flowers.. But I know, the look is so - irregular -
I think if you cut down all the weeds to bare ground, rake as much out as possible, aerate if possible, put down your fertilizer, compost, whatever you decide to use, and the bermuda seeds, it will come back and it being so aggressive, it will take over the place and choke out the weeds.. It can also grow really thick and high. It is pretty much indestructible when its taken care of..
One more thing that can help with weeds - After - you get your lawn up and going great - a pre-emergent herbicide, applied in early Spring before any weed seeds germinate.. It will leave a layer on the soil that will kill most seed growth as soon as they pop up and keep them from fully growing for so many months..
You just have to remember this - a pre-emergent herbicide will kill - All - seeds, bad ones and good ones.. There is no "smart" pre-emergent herbicide out there... All or nothing..
But you have to water the lawn.. Or anything else you plant and want to see again, unless you have regular rain from above...
My family in Southern Texas ( San Antonio), all have a grass called St. Augustine.. It grows by runners on top of the soil.. It can be finicky for them down there in 90% humidity day and night weather, etc..But it is what everyone uses, except some that go with Bermuda grasses..
What kind of soil do you have there??? Hardpan, Lake Bottom Clay,,Caliche?, Sandy? Sandy Loam? All these things have to be looked at to be able to give you the best advice..
I was stationed in Northern Texas for a spell once.. Sweetwater AFS, Sweetwater Texas.. Abilene was close by and they had a huge SAC (Strategic Air Command) airbase there.. Those were the days...
Good luck with this..