Answer: The conditions you describe will make it difficult if not impossible for you to have a lush lawn. Unless you want to remove some of the trees, or at least lower limbs, to let in more sunlight, I'm afraid grass simply won't thrive. Would you consider some other types of ground covers? There are some, like wild ginger, bugleweed, English ivy, pachysandra or myrtle, that should do well in those conditions. They won't tolerate foot traffic, however. If you want to be able to walk in the area, could you construct flagstone walkways and interplant with the various groundcovers?
Before you do anything, consider having your soil tested. Oak trees can indicate an acid soil, so I'd at least test the pH. And resist the temptation to add a thick layer of new topsoil over your clay soil. This could smother shallow tree roots.
Hope this helps!
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