This is an old post, i know, but still relevant to people reading it for information.
One of the best things you can do for dense clay soil is let the weeds grow tall then hand scythe them (less compacting than mowing) then leave the stalks as a mulch
This takes a year or two, but there's a reason behind it. Most of what we call "Weeds" have long, deep taproots designed to break up the soil and draw nutrients from deep down up to the surface.
If you let them develop these deep roots then cut them, the roots mine nutrients from below, rot in the broken up soil, and the tops
compost the nutrients back into the upper layer.
What you end up with is way better soil than you had.
Comfrey, alfalfa, and yellow clover are great for this for many reasons you can read up on.
But common dandelion, chicory, thistle, red clover, etc will do the same.
Afterwards, you can simply add more mulch over them to kill them off wherever you want to garden.