The actual source is from non-Jewish naturalists, dating in some for back to Aristotle, who also believed in some sense that man is in a different category (teleologically, and therefore categorically). It was later refined into a rigorous taxonomic system.
The idea was picked up by Jewish scholars from the Medieval Era, though some of them use the word מרגיש instead of חי to refer to animal life. This includes the Kuzari (1:31-39), by the Radak (in his comments to Yirmiyahu 10:8), Rabeinu Bachaye (to Shemos 34:35 and Kad HaKemach, Pesach 1), R. Yehoshua ibn Shueib (Parshas Emor), R. Yosef Albo (Sefer HaIkkarim 2:30), Abarbanel (Parshas Pinchas ch. 28), and is hinted to in the Meiri to Avodah Zara 42b. After this period, it becomes extremely widespread in Jewish thought, appearing almost a dozen times in the writings of the Shelah, for example