How do the clues at a crime scene of a mysterious death investigators determine what might have occurred and identify or exonerate potential suspects?

Well if there was a murder weapon there at the death scene, the investigators check the weapon for fingerprints. Same for if there are random murder objects lying around anywhere, they'd check it for fingerprints. I suppose after checking the weapon or murder object, they set it under a scanner that is connected to a computer and it tracks who's fingerprints were last on the murder object and the title and picture of that person comes up.
Hope this helped.