Picking a Herbicide
There are a number of herbicides that will kill turf grass. Most are not recommended by horticulture experts, as they leave a residual in the soil.
Other choices, such as glyphosate (the active ingredient in Roundup™), do not leave a residual in the soil. The spray is absorbed by the plant and root system. Products like Roundup can be very expensive and are best used in small areas or to spot spray, if needed.
Always follow the directions and use appropriate protective gear when using herbicides.
Once the turfgrass is dead, you need to add compost/mulch to compost the dead lawn in place.