Like Elfie said, till it up, it may be a clay and compaction issue.
And the tree species has a lot to do with this, if it is a root runner, it can layout a web of roots that can cause issues, or if it's an oak, it will kill everything that tries to grow under it, depending on the variety of oak.
Also pine, Cedar and Redwoods are highly acidic, and not much grows under the drip area of the needles from rain.
If the tree isn't all that important, cut it down, till up the soil, and lay some sod.
Or pave it and make a tennis court. ;)
I took horticulture in college for a year, and the one thing I learned was, look at what lived there before man ever arrived, and those are the plants you should consider, and that goes for grass as well.
You can't force Mother Nature to bend to your needs.
That's why we don't grow cherries or pear etc, the soil is not conducive to supporting certain kinds of plants, not to mention weather.
Elfie is right, there are some grasses that grow better in the shade, but they usually don't tolerate traffic.
Post a pic if you can, I'm sure someone here has an eye for decorating, God knows I don't. ;D