This question is hard to answer without knowing a bit more about what conditions run in your family and what, specifically, you are interested in testing for.
Generally speaking, most chronic conditions that manifest later in life don't have any specific test that can be performed in childhood. However, it is recommended, for example, that children who have a strong family history of high cholesterol be tested at least once early in life, and that is a simple blood test that can be performed. Similarly, if there is a strong family history of diabetes and if your child suffers from being overweight, testing for diabetes is another simple test that can be performed.
On the other end of the spectrum, many rare genetic conditions do have specialized tests that can be performed in family members, including children. The exact details of the testing and its availability would depend on the specifics of what condition we were talking about and would probably require a consultation with a genetics specialist.
At the end of the day, the most important thing you can do to make sure your child stays healthy is to make sure that he has regular checkups with his pediatrician