Break the clay into small pieces with a shovel, trowel or even the tiller. The pieces should wind up smaller than the size of pea gravel. Clay left in large chunks will not improve the soil.
Till the soil to a depth of about 24 inches. Most plants do not grow further into the soil than this. If you want to grow trees in this location, however, then till at least 3 feet deep.
Remove about 1/3 of the sandy soil with a shovel. Haul off the sand with a wheelbarrow.
Create a mixture of equal parts clay, organic compost (any mixture is fine), humus and black soil. You can use potting soil or move soil from a nutrient-rich bed to the sandy bed. This mixture should equal the amount of soil removed from the bed.
Spread the clay mixture over the bed with a shovel and rake. Till the two soil types together until they start to blend seamlessly as one soil type.
Add a general fertilizer over the soil. Use a slow-release fertilizer that releases nutrients over a period of several months. Water the soil to settle the dirt and complete the mixing process.