Before you dump a bunch of mulch over your layer of clay, you need to address the site drainage. Since clay blocks the water movement downward, it will either flow sideways or form puddles in low areas. Make sure you grade the area so that water flows in one direction away from any buildings.
It is important to choose the right type of mulch for amending clay soil. Mulch with large particles like wood chips will rob the nitrogen from any plants as it uses the mineral to break down. This type of mulch also takes the longest to break down into useable particles for plants. Choose mulch that is very finely ground such as root mulch or even humus.
Since you want to break up the clay, you will need to rototill the mulch into the layer of clay. Dig a shovelful straight down and measure the layer of clay. Add a corresponding amount of mulch for the depth of clay. For instance, for a 2-inch layer of clay, add 2 inches of organic material. You can add up to 6 inches of mulch, since the grass will not reach down much farther than that.
For areas where breaking up the layer of clay is not feasible, you can drop a 6-to-8-inch layer of good quality screened top soil over the area. Make sure it is full of organic material to hold moisture, adding peat moss if necessary. This layer should hold enough moisture for the grass roots. Check the pH to make sure the range of the soil is between 6 and 7 for a healthy lawn, adding amendments as necessary before laying the sod.