Zebra - my garden has heavy clay, as does my allotment - but nothing compared to the garden at my old house, which was down the road from a clay pit. It stuck together so much that you had to dig it out in breeze block squares and then break it down. shudder
I think if you try to break it up dry, you will end up damaging both yourself and the soil! A bit of rain will loosen the lumps naturally. This is not technical knowledge or anything, so I could be wrong about it, but here is what I do: I mulch with a really good, thick amount of cover (manure/compost/spent mushroom compost) over the surface and that seems to lock the moisture in a bit, and encourages the worms to get in there and do some of the work for you! You need absolutely bags and bags and bags of the stuff, though. I have loads of old Ikea bags - the blue ones - and I take them across town to Poo Mountain (near a riding stables) and dig out loads and loads, shove them in the back of my ancient and knackered Polo, and then spread everywhere. Because it's well-rotted, it doesn't smell. Adding a bit of horticultural grit helps a lot, as does gypsum.
It is a battle you have to fight over and over again though- have a couple of bits of my garden that are really wet and I must have dumped dozens of bags of manure, grit, and compost on them and they are still not there, but improving.