Weedkiller near my vege garden

If you are choosing not to go organic you could use broad spectrum, non-selective herbicide such as Glyphosate 360, wait for a couple of weeks and do again and remove remains of lawn, then prepare the soil and plant. If you are looking for an organic method you can cover the lawn with black plastic or old carpet for a few weeks/months, this will slowly kill the grass off then I would suggest you raise the garden using the ‘no dig garden method’. Basically it is gardening inside a big-sandbox type frame of wood or other permanent material and filling it with good soil/compost.