Your county extension office will typically perform a soil test based on your samples. The results will contain specific instructions for improving the quality of the soil, including how much lime to add, if necessary. As far as lime goes, a soil test is also necessary because it will tell you the "magnesium base saturation" percentage and determine whether you need calcitic or dolomitic lime.
Clay soils with a magnesium base saturation over 20 percent should be amended with calcitic lime. According to Jessica Walliser of "The Pittsburgh Tribune-Review," calcitic lime is recommended for soils with high levels of magnesium, while dolomitic lime is recommended for soils suffering from magnesium deficiency. Since clay soils don't often suffer from magnesium deficiency, calcitic lime is often recommended.
Amendments are necessary because clay soils tend to hold moisture too well and are also quickly and easily compacted, especially during wet weather. It's difficult for plant roots to move through clay soils if they're very compact, and amendments work to separate the clay particles, making it easier for water and roots to pass through the soil. Lime is one amendment for clay soil, provided it's also necessary to raise the pH, thereby making it less acidic. Compost, which most people can make at home for free, is just as effective. You can also amend the soil with coarse sand--also known as builder's sand--in addition to compost.
It's best to amend large areas of soil at a time, rather than small areas that will continue to restrict root growth as the roots try to grow beyond the amended area. This may seem overwhelming, especially if you have a large area to work with. Working with one small area per season is a perfectly acceptable approach. A rototiller will get the job done more quickly than working by hand.