How to clear an overgrown flower bed and prep for grass
I had a similar issue in my back yard (the whole thing). It was just weeds. I killed all the weeds with a dose or two of Roundup and had the entire 7000+ sqft tilled up. I then raked it flat and seeded most of it. It now has less weeds than my front lawn which received sod prior to moving in. This was new construction, and the sod was part of the package.

If I had to do just a flowerbed, or the whole yard again, I'd vote for doing it the same way again.

Just be sure to wait a week or two after applying the Roundup so it works it's magic and the weeds are dried up completely. This also aids in tilling it, by hand or machine. The Roundup should not last long on barren soil and will not affect the seed after a couple of weeks.