The phenomenon of inbreeding increases the level of homozygotes for autosomal recessive genetic disorders and generally leads to a decreased fitness of a population known as inbreeding depression which is a major objective in clinical studies.
I think that the solution to this very real problem is counter intuitive to our cultural and societal norm/beliefs that we should marry and have children with those from similar cultural, religious, and ethnic backgrounds. In the animal kingdom, males often leave the family unit and breed with females in other groups. This adds strength to the species. Even though humans are not generally having children with immediate relatives, they tend to stay within their familiar group for social reasons. This is why we get thalasemia, tay saks, sickle cell, and other varied cancers and metabolic diseases. Our social practices have not caught up with our genetic needs.
If you want to measure the effect of inbreeding on increasing a certain disease, you will have to obtain a measure of relatedness between the individuals in the population. You can do this through genetic markers from a large enough representative sample of the population you study. Of course you will have to measure the disease and then compare the prevalence of the disease to some other situation. For example, you measure one highly inbred population and one less inbred population. Another possibility, is to measure the individual risk of having recessive autosomal disease and how that risk is affected by the degree to which an individual is highly inbred. A final possibility is to use proxy measures, for example marriage records over several generations and the number of early deaths in the offspring. This approach could be fruitful in situations were you have extensive records over several generations and early deaths are "truly" indicating uncommon diseases. Such data, I assume, are difficult to obtain. I recommend Alan Bittles studies as a starting point.