See if you can find someone locally who has horses and will let you have some manure and also go to the council and get a couple of compost bins. Use one bin for the manure as this will then turn to compost reasonably quickly. The other for some grass cuttings, cardboard, kitchen waste etc.
This you can dig into the patch of ground you are working on once it is compost.
Also I would say if you want ground for veg growing quickly I would recommend raised flower beds or large black veg bed bags that you can put down, it'll kill off the grass and then once you've grown something you can spend the time on the ground underneath.