The first Christian author who made the distinction between heaven vs. sky appears to be none other than the apostle Paul.
At 2 Corinthians 12:2 (ESV) we read:
I know a man in Christ who fourteen years ago was caught up to the third heaven - whether in the body or out of the body I do not know, God knows.
The context indicates that Paul was presenting his credentials as a true apostle, so the statement is likely made about himself even though he refers to the "man in Christ" in the third person. But the point of interest here is the mention of a "third heaven," indicating a belief in multiple heavens. The lowest heaven would be that observable from the earth, and the highest heaven would be God's dwelling place.
We don't know exactly what Paul meant by the "third heaven," as the book of 2 Enoch (or Slavonic Enoch, not generally considered a Christian writing) refers to Enoch's traveling through seven heavens to reach the presence of God. (2 Enoch 3-37) Was Paul's third heaven the third of seven, or did Paul consider it the highest heaven?
In turn, the belief in multiple heavens may go back to a prayer attributed to King Solomon at the dedication of the temple. According to 2 Chronicles 6:18 (ESV), Solomon prayed:
But will God indeed dwell with man on the earth? Behold, heaven and the highest heaven cannot contain you, how much less this house that I have built!
The second century Christian writer, Justin Martyr, referred to God as dwelling in "super-celestial realms," indicating that God dwelt beyond the visible heavens. He wrote regarding God the Father:
. . . no one with even the slightest intelligence would dare to assert that the Creator and Father of all things left His super-celestial realms to make Himself visible in a little spot on earth. (1)
The idea that God dwells in a literal place called "heaven" seems to conflict with another Christian belief, God's omnipresence. If God is present everywhere, how can he be confined to a particular place? Augustine of Hippo attempted to solve this problem by stating that the redeemed, even though they would enter heaven, would not behold a localized deity but would discern God's presence everywhere:
Wherefore it may very well be, and it is thoroughly credible, that we shall in the future world see the material forms of the new heavens and the new earth in such a way that we shall most distinctly recognize God everywhere present and governing all things, material as well as spiritual . . . (2)
With our expanding knowledge of the universe, "heaven" may be a symbol of God's transcendence, and refer to a state of being rather than a place.
(1) Dialogue with Trypho. Chapter 60.
(2) The City of God. Book XXII:27.