This is easy to fix and not very costly. Clay soil is a soil that (a) lacks calcium (b) is compacted through weight (c) Has poor structure due to to high cation exchange capacity and low organic matter. Or a combination of the three.
Firstly, DO NOT REMOVE ANY soil. And DO NOT ADD any new "soil". As stated before, clay soil simply has bad structural properties because its missing something, but are that can be easily rectified. So we NEVER remove/add soil, you just add the components your soil needs.
For your first step, The grass has to go for now. Mow the grass very low, then dig the remaining grass through your soil using a hoe or shovel. Secondly is calcium (in the form of gypsum). You can test if your soil is low in calcium by placing a small chunk of soil in distilled water in a jar, if after a few hours the water around the soil is murky or cloudy, you will need to add gypsum. Thirdly, weather you have added gypsum or not, add large amounts or organic matter, like compost or in bags from a nursery. Mix this organic matter through the top 5 - 8 inches of soil. Also if you have access to the product "Multicrop GroundBreaker". I would recommend it. It is a liquid form of clay breaker, if you can get it, use that as your last step. Your soil should have good drainage after a couple of weeks. Finally, sow seeds of your new grass.
In closing, improving your soil will ALWAYS produce better results then removing and adding new soil.