There's two ways to do this; which one you pick depends on the effort you're willing to put in, and how much time / money you have.
- Add at least 6" of compost, and till up the lawn. [You can do this in sections over a number of years for a lessening of the total task in any given year.] Reseed, and mist daily until you have new growth well-established. Then, go to regular lawn care.
- Add a 1-2" top dressing of compost spring and fall. Overseed bare patches. In the due course of time, the topsoil will become more loamy than clay.
If you provide more organic matter, earthworms will be likely to migrate to your yard naturally.
Chris Owens