NY Times food writer Mark Bittman has suggested that if you don't have your own grinder, either stand-alone or attached to a KitchenAid-type mixer, a standard food processor can do a fine job of meat grinding, if you watch what's happening:
Next, don't overprocess. You want the equivalent of chopped meat, not a meat purée. The finer you grind the meat, the more likely you are to pack it together too tightly, which will make the burger tough.
If you want to delve a bit deeper into the specifics of ground beef issues, a Grocery Guy blog post will indulge your curiosity—just don't read it right before lunch. That said, he brings up a nice halfway compromise to making a mess of your kitchen counter: Get to know your butcher, and have him or her grind your meat to order. That post also contains a few more nitty-gritty tips on cuts, seasoning, and patty preparation.