Many laboratories now use liquid scintillation counters with the samples being converted to benzene.
All of these counter types measure the C-14 content by monitering the rate of decay per unit time.
During the lifetime of an organism, the amount of c14 in the tissues remains at an equilibrium since the loss (through radioactive decay) is balanced by the gain (through uptake via photosynthesis or consumption of organically fixed carbon).
However, when the organism dies, the amount of c14 declines such that the longer the time since death the lower the levels of c14 in organic tissue.
Working with several collaboraters, Libby established the natural occurrence of radiocarbon by detecting its radioactivity in methane from the Baltimore sewer.