Are all diagnostic tests for klinefelter's syndrome done locally? What are they?

Blood test. Most localities have access to chromosomal analysis. Usually, labs send these blood tests out to a reference lab to be analyzed. So the blood can be drawn just about anywhere.
Chromosome analysis. Klinefelter's syndrome is a condition where a male (genotype xy) has an extra (maybe even several) x chromosome (xxy). The test is a karyotype, which is simply to pull all of the chromosomes out of a cell, line them up, identify them and count them. Whether this is done locally or not depends on where you are and what facilities are available. See your doc for more information.