score:5
Let's break it down to two different parts:
Wikipedia info on C.J Thomsen.
So, the short answer is relative-dating, i.e. "the depth of archaeological finds" in the question. Carbon dating is absolute dating which didn't exist then.
On the Method of Seriation:
CJ Thomsen used seriation. From A Dictionary of Archaeology, p.520 - emphasis mine:
Christian Jurgensen Thomsen (1788-1865) was the earliest exponent of seriation. In 1816, he began to use the method to organize the collection of pre-historic artefacts in the National Museum of Antiquities at Cophenhagen ...
On Classification into Three-Age System:
CJ Thomsen was the first to propose the three-age system in 1836 (other sources point to 1819). He was an archaeologist, specialising in numismatics and developed the tripartite system according to the material (artifacts). As explained, this is relative-dating which resulted in culture and artifact classification.
The date of 1819 is provided here - Encyclopedia of Archaeology (2007), p. 1020 - emphasis mine:
It was however Christian Thomsen who first gave wide publicity to what has come to be called the three-age system, when in 1819 he arranged all of the prehistoric collections in the newly opened Danish National Museum into separate Stone Age, Bronze Age, and Iron Age assemblages. The three-age system was from the beginning both a culture classification and a kind of artifact classification.
However, I believe the correct date should be later, closer to 1836 because:
the tri-partite division of the past that had existed ever since C. J. Thomsen, in 1836, had ordered the collections of the National Museum in Copenhagen into cabinets containing stone, bronze and iron objects.