Grass Dying Off
The best remedy for improving clay soil is to add gypsum which breaks up the structure a little and allows better drainage. Clay soil is rich in nutrients, however if it is not properly aerated and/or overwatered, problems can occur with poor drainage and air starved grass roots.
Good aeration is vital for a healthy lawn. I would suggest that you thoroughly aerate the affected area with a garden fork and then add some gypsum and peatmoss to prevent the soil from closing in again.
Any new grass seed sown needs to be kept moist for the first couple of weeks, then water less frequently in order to establish strong root growth.