Trying to grow anything in clay soil without improving it won't get you very far. Clay soil suffers primarily from being too dense for strong root systems to grow and penetrate--not to mention get water. Clay soil retains water very well but has a difficult time moving. With a little amendment, however, it's not difficult to turn clay into growing soil, which can yield a plentiful harvest.
Use your soil pH test kit by following the instructions. The kit will tell you about the chemical composition of your soil and help you to determine what type of amendments or compost are needed. Add lime and gypsum if your soil needs calcium. Bone meal can help phosphorous depletion, while sulphur is a good option if the soil is particularly alkaline.
Rip up the top 4 to 6 inches of your soil using your chosen digging tool. Break it up as small as you possibly can. This may take some effort as clay soil is notorious for coming up in big chunks when digging into it. Be patient and break it up; your garden will benefit later.
Add compost and amendments recommended from the soil test results. Follow the compost's directions for application or, if you have homemade compost, follow your recipe's instructions.
Plant your garden at least a week after you have amended your soil. Planting sooner risks harming the tender, young roots of new plants.
Repeat this process every growing season. You may even wish to try this process right before winter so everything is ready to go when the first thaw occurs and you prepare to get new plants into the ground.