Horticultural vinegar is a good green choice to kill weeds without dangerous chemicals. It kills young weeds and is stronger than the kitchen vinegar so use it carefully. The weeds will be dead within two to three hours. Though your garden may smell like a salad you will be using a friendly garden pest killer.
You must also make a 3 to 4 inch thick layer of organic mulch, such as shredded bark or wood chips which will help to keep weeds out and weaken the ones that are still troubling you. Green Your Home All-in-One for Dummies also shows you how sheet mulching is done to murder your weeds! Pull out any surface weeds, cover your soil with two or three layers of corrugated cardboard salvaged from waste, overlap the cardboard and wet it down firmly, and then cover the whole contraption with around 4 inches of organic mulch, such as wood chips.
Leave it there for a few months and you will be rewarded with your weeds being killed much more effectively than herbicides. It will be the cheaper and the safer choice.