When did the concept of οὐσία begin to refer to God's nature? In the NT, the Greek word οὐσία means "possession, property (substance)." To simply put, it is a thing or stuff that someone owns. 


  And the younger of them said to his father, ‘Father, give me the share
  of property that is coming to me.’ And he divided his property
  (ousia) between them. Luke 15:12 (ESV)


There is no NT passage that relates οὐσία to God's nature. Even writings that predated the NT did not use the term to refer to God's nature.


  He treated his parents-in-law with great respect in their old age, and
  buried them in Ecbatana of Media. He inherited both the property (ousia) of
  Raguel and that of his father Tobit. (Tobit 14:13 NRSV).



Concpetually, I do hold onto the applicability of οὐσία to refer to God's nature since nature is what innately belongs to someone.
I agree that the Greek word homoousios (same substance) faithfully preserve the meaning of Scripture in creedal form (Nicene Creed cf. John 1:1, 18 10:28-33; Hebrews 1:3).


When did the concept of οὐσία begin to refer to God's nature?


Who is the first church writer /church leader that refers to οὐσία as God's nature? 
Why is οὐσία employed? What does other Greek words don't have that only οὐσία could give?
The earliest use of οὐσία to mean the substance or essence of a thing is by Aristotle in his Κατηγορίαι, though Aristotle attributes its earlier use to Plato. Justin Martyr comments on Aristotle's description of the nature of the Divine, confirming that Aristotle (along with Plato) uses the word in the manner described as early as the 4th century BC. So it appears that the first use was not within the Church, but within Greek philosophy and that οὐσία was used by the Greek Fathers in this manner following the philosophers. This so-called "Pagan" origin had an effect on the later use of the word within the Church. 

Tatian and Justin Martyr use the word as early as the second century to discuss the substance and nature of God in the context of the Logos. Consider the following passage from The Holy Trinity: God for God and God for Us, by Chung-Hyun Baik.


  Both Justyn Martyr and Tatian of Assyria went further to hint that the Logos has the same ousia as God the Father. Justin Martyr stated that the Logos, which is invidisible and inseperable from God the Father, was generated from the Father by the power and will of God the Father,'but not by abcission, as if the ousia of the Father were divided' (Dialogue with Trypho, 128). Justin Martyr provided an analogy of human speech, by which it is meant that, though we beget and utter a word, our power of uttering words would not be diminished. He offered another analogy of a fire kindled from a fire; an enkindled fire is distinct from the original fire, but the original fire remains the same undeiminished fire even after it ignites another. Tatian of Assyria stated that the Logos was in God the Foather and came into being 'by participation, not by abcission' (Address to the Greeks, 5). For him, abcission implies a separation of the Logos from the original ousia of God, while participation presupposes no deficiency of the Logos in the original ousia of God. He illustrated this point by the analogy of a torch. Though one torch lights another fire, nevertheless the light of the first torch is not lessened by lighting another fire (Ibid.)"


It is well known that Justin read Aristotle, and in fact uses οὐσία in his comments on Plato and Aristotle in Horatory Address to the Greeks (Ch. 5), 


  "For Plato, with the air of one that has descended from above, and has accurately ascertained and seen all that is in heaven, says that the most high God exists in a fiery οὐσία. But Aristotle, in a book addressed to Alexander of Macedon, giving a compendious explanation of his own philosophy, clearly and manifestly overthrows the opinion of Plato, saying that God does not exist in a fiery οὐσία: but inventing, as a fifth1 οὐσία , some kind of aetherial and unchangeable body, says that God exists in it. Thus, at least, he wrote: 'Not, as some of those who have erred regarding the Deity say, that God exists in a fiery οὐσία."


So, οὐσία was used at least as early as the second century by the Greek Fathers, though the use of the word in the manner in question likely follows from Aristotle and therefore predates the birth of Christ and its use within Christianity. 



1. Note this "fifth essence"- the first four being fire, water, air, and earth- is the origin of the English quintessence, as well as the identity of the lapis philosophorum, the "philosopher's stone" about which the alchemists wrote, and whom the Christian alchemists identify with the transfigured body of Christ.