What is the best way to clear an overgrown garden with mostly grass and weeds without using chemicals?
I'm not sure if your garden has plants in it you wish to save (as in a badly neglected once-nice garden or if you plan on redoing the entire plot.
In the first instance, I'd dig up plants you want saved and lay the root balls in a freshly dug trench (the plant leaning on one side of the trench) in the shade, water well, loosely toss soil over the roots and water again. They'll be fine there until your plot is ready for replanting. Be prepared for some set back if the plant is blooming or about to though. Still, it'll be good the next year. Then proceed with clearing the soil of weeds.
Tilling or digging up the soil, then removing weeds and roots is good but it won't prevent weed seeds that have laid dormant in the soil from sprouting all over the place now that they have a chance to fresh ground and sunlight. If you don't mind weeding or hoeing afterward by hand, the suggestions given are suitable.
If you'd rather have clean ready soil, there is another method that's similar. It's called Soil solarization
Basically, it's clear plastic tarp placed over the soil and weeds. It uses the heat from the sun to kill weeds, their seeds, insect and slug eggs, and soil pathogens. Clear plastic (as opposed to black plastic) allows the heat to penetrate much deeper. This site explains how to do it effectively and even if the weather isn't hot yet, how to modify the method to make it just as effective.