How does one tell (or conclude) if a substance is carcinogenic?
How do we tell or suspect one compound to be carcinogenic?

As written in the comments to the question, this the result of large studies on the human population, correlating blood or urine levels of some substance to instances of cancer. Alternatively or concurrently, studies on animal models, such as mice, rats or even in-vitro studies may be carried out. As a result, regulatory bodies may include a substance in lists, which may preclude them from use in consumer products, manufacturing procedures etc.

Are all carcinogens organic compounds?

No, this does not have to be the case: plain arsenic and basically all inorganic cadmium compounds have been classified as suspected or definitively carcinogenic.