I've seen this many times in new subdivisions. The worst was a house with heavy clay soil where a year after the builder left you could still roll up the grass. As you have found grass finds it hard to get roots into a compacted clay subsoil.
I assume that you are not able to remove the existing grass and add two to six inches of topsoil. That would solve your problem immediately but would change the drainage around your house and that's never a good thing.
Stopping the grass from browning out in the summer when water loss due to heat is more than the roots can take up from the soil could be as simple as watering once a week during the dry periods.
A longer term solution is to sow more grass seed and top dress twice a year, spring and fall, with compost to a depth of 1/4" to 1/2".
Another solution that is hard to back out of is to plant clover and thyme. Clover will break up the clay and add nitrogen but cannot be easily removed once added to a lawn.
As far as grass mixtures for most homeowners the choices are "Shady lawn" or "All purpose".
The only difference is the ratio of different grass species. For real success you need to improve the underlying soil.
Good mowing practices help too. Cut late in the spring and cut as high as you can stand to let the roots store energy. Continue cutting higher in the back yard during the summer and then cut lower in the fall to avoid snow mould if that happens in your zone.