Speaking to the killing of the grass only, I know from Landscaping projects that you can put down layers of newspaper over that grass and it will eventually die, but the fastest and most simple procedure is using something like Roundup.

Removing grass by digging it out might not be successful anyway, because you may end up leaving the root system intact. Bermuda Grass, especially is difficult. So, my layman's opinion is to use something like Roundup on the areas you want to cover, and proceed with your build as planned.