Right - on that plot I would...
Not bother to dig out the grass, weeds or dandelions just yet. Try and get it level or mark out the paths and beds...I'd cover the whole lot with cardboard and buy some straw bales and put over the cardboard. It will help weigh it down [you can use some of the bricks for this too]. And as they rot, will help to add organic matter to your clay. It also stops the clay drying out which it will do the moment the sun comes out. If you are going for paths and beds, I'd get some wood chip and put that down for the paths to walk on for now.
When you are ready to start growing things, cut a cross in the cardboard and peel the cardboard back, plant the plant [eg potato, squash, corn] in a hole dug under where the cardboard was, pop the soil around, water in well, and then put the cardboard back around the plant. It will help to keep the weeds down, kills the grass, soak up the moisture and stop it evaporating and leaving you with a clay cap which is rock solid and which also constricts the plant as it grows.
As the cardboard rots, then start to dig out the weeds that grow through. A fair amount will die due to their light being stopped, and you can dig the roots out at a later date. If you carry on doing stuff to that as is, you are going to squeeze out all the air from the soil, making it worse in the long run. At least with cardboard on it, as you walk on it, it distributes the weight across the soil.
As you harvest later in the year and clear small areas, then is the time to dig out the roots, and dig in some sand and well rotted manure, and again mulch over winter. This time you could use cardboard, straw, or sow a green manure that keeps a plant on the soil over the winter, and in the spring you chop it down and dig it in around a fortnight before you plant up next year's crops.