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You have a two good options. 1) If you want to go organic, just smother everything with something dense -- a thick layer of leaves, cardboard, old carpet newspaper, black plastic, whatever. If you find it too ugly (which is likely) cover whatever you use with a thin layer of mulch such as wood chips or pine needles. Depending on what kind of lawn you have, it can take a few months for everything to die (aggressive southern grasses such as St Augustine take the longest).

2) Spray it with RoundUp. Wait about 10 days to see how much survive the first treatment, and spray again. You might need 3-4 treatments.

What you don't want to do is till it up with a rototiller, because that chops up plant parts that then sprout new plants, and also brings buried weed seeds to the surface wher they regrow.