Gatsby's "platonic conception of himself" most likely refers to Plato's idea that "truth is an abstraction." Simply put, Plato argued that ideas are separate from reality.
Gatsby's idea of himself as someone "great" did not match the truth of his reality, which is that he was the son of poor farmers.
You can think this:
James Gatz (reality) - The son of a poor farmer
Jay Gatsby (abstract) - Gatz's ideal of himself, which he creates from his own mind.
The interesting line in the paragraph that talks about Gatsby's "platonic conception of himself" is the last sentence, which says, "So he invented just the sort of Jay Gatsby that a seventeen-year-old boy would be likely to invent, and to this conception he was faithful to the end."
Here, Nick might be suggesting that Gatsby's platonic conception of himself is immature and this, perhaps, is what leads to his death. This immature conception of himself leads to Gatsby's belief that he can relive the past or that Daisy will leave the security of Tom or that his protection of Daisy after she kills Myrtle is somehow the right thing to do to help him become great.
Platonic conception is an idealism by Plato that every being on Earth aspires to some perfect or vastly superior form of itself. Fitzgerald uses this concept to characterize the very embodiment of Gatsby. James Gatz was the son of two "unsuccessful farm people" (78) who had no social standing or money. "So he invented just the sort of Jay Gatsby a seventeen year old would be likely to invent, and to this conception he was faithful to the end." (79) Nick is telling us James Gatz sacrificed his character, his ethics, and ultimately his identity to become Jay Gatsby. However, when Gatsby met Daisy, his platonic conception of himself aspired to new heights. Becoming wealthy was no longer enough, Gatsby was filled with an obsession driven by both love and a desire for social position to have Daisy. Gatsby's dream would never be complete without her. The Jay Gatsby we read about, the wealthy charming young man besotted with Daisy, "spr[u]ng from his Platonic conception of himself." (78) Ultimately the author is trying to tell us we must act upon our dreams in order to realize them. The American Dream is only within reach if you reach for it.