The contribution of genetics to human disease can range from 0% to 100%. There are examples of diseases that are basically 100% genetic; if you carry the gene for the disease, you will have the disease. Jim has listed some examples. There are also examples of disease that have almost no genetic component. For example, many infectious diseases—like HIV, influenza, and chicken pox—are caused by pathogens from the environment.
Of course, not everything in biology is so simple. The environment can change how genes are expressed and genes can affect how we respond to the environment, creating many cases in the middle. For example, some purely genetic diseases can be ameliorated by changes to an individual's environment. For example,
phenylketonuria (PKU) is a genetic disease characterized by the lack of an enzyme needed to metabolize the amino acid phenylalanine. PKU can lead to neurological problems, but these neurological effects can be avoided if the individual avoids eating foods with high levels of phenylalanine. Genetics can also influence diseases caused by environmental factors like pathogens; for example, some individuals contain mutations that give them resistance to infection by viruses like HIV.
There are a wide variety of diseases and disorders—cancer, Alzheimer's disease, cardiovascular disease, autism,
partisanship, etc.—that have some genetic component and some environmental component. To estimate the relative contributions of genetics to these diseases, researchers will rely on a number of different types of studies.
One type of study involves looking at the rates of the disease in identical (monozygotic, MZ) versus fraternal (dizygotic, DZ) twins. Twins are generally raised in similar environments, but MZ twins will share 100% of their DNA while DZ twins only share 50% of their DNA. Researchers will look at the rate in which the twins are concordant for the disease; that is, whether they both have the disease or both do not have the disease. If the rate of concordance is higher in MZ twins than DZ twins, then the disease has a genetic component, and the difference in the rate of concordance between MZ and DZ twins can allow researchers to estimate roughly how much genetics contributes to the disease.
To identify genes that contribute to specific diseases, researchers can perform
genome-wide association studies, in which they identify people with a particular disease and a similar group of people who lack the disease. They can then compare the DNA of the two populations to identify genetic markers associated with the disease. Follow up studies can then identify the genes near those markers and determine how they contribute to the specific disease.
Further reading (aimed at a general audience):
https://ghr.nlm.nih.gov/primer/inheritance/heritability
For a comprehensive, technical discussion of the topic, see:
https://www.nature.com/articles/nrg3377