The way Cavendish did it would seem to be simplest - it can be done with C1818th Century technology.
In practice its very difficult to do in a lab, alternately it is a very good way to demonstrate the resonant properties of concrete lab buildings.
If you wanted to try it for yourself then a garden shed, with an insulated inner room/box built of plywood and styrofoam and a vibration proof table made from stacks of inner tubes and paving slabs, and a telescope+webcam so you can keep it completely sealed. Seems plausible.