The neoclassical period of literature is said to have emerged after the Restoration, roughly beginning in 1660, and this is said to have carried on until the emergence of Romanticism, which many argue to have begun with the publication of Lyrical Ballads in 1798. It is very difficult to argue that there is a single definition of neoclassicism, however, generally speaking, neoclassical authors believes in some of the following tenets: a strong traditional approach coupled with a reverence for classical writers from Ancient Greece and Rome, literature viewed as an "art," poetry was meant to be an imitation of human life and the main subject of literature was meant to be human beings and lastly literature was to explore the characteristics of human beings.
"The Vanity of Human Wishes" can be seen to be an excellent example of neoclassical literature in the way that its strict poetic form obviously pays homage to Classical writers. Remember Johnson wrote this in imitation of the Tenth Satire of Juvenal. In addition, note how this text certainly presents the realities of human life and the characteristics of human beings are explored in all their folly and vanity through examining the centrality of wealth and gold and how humans crave it in spite of how much damage it causes:
For Gold his Sword the Hireling Ruffian draws,
For Gold the hireling Judge distorts the Laws;
Wealth heap'd on Wealth, nor Truth nor Safety buys,
The Dangers gather as the Treasures rise.
As such, this text certainly does act as a "mirror held up to nature," which was a common phrase used by neoclassical authors to describe what literature should do in reflecting humanity. This text, through its focus on the characteristics and exploration of what it means to be human, coupled with its Classical form, certainly seems to be very representative of neoclassical literature.