Jesse E. answered 06/04/19
Masters in Chemistry and Bachelors in Biology
Carbon is introduced into living organisms during reproduction, whether it be sexual reproduction( as with mammals, reptiles, and fish) asexual reproduction(as with plants), or binary fission (as with bacteria). Once fertilization occurs with asexual or sexual reproduction, or a new formed cell with binary fission, a cell has the carbon that it needs to begin developing into an adult individual or cell.
As for carbon was first introduced into living organism, that is a loaded and polarizing question that depends on your view of religion.
As for how carbon continues to be introduced into living organisms, that occurs by food-intake with animals and photosynthesis in plants.
P.S. Given the variety of animals and other living organisms, this is an over simplified answer to the question. There is more to be discussed but it is beyond the scope of this format.