No, there's no idiom here or at least there is not as far as I am aware. I think your problem is that the Oxford Pocket Dictionary's definition is too brusque. I believe this definition of Swagger will help you understand what is actually meant much better:
To bluster; to bully; to boast or brag noisily; to be tumultuously proud.
The American Dictionary of the English Language (A.D.E.L.) by Noah Webster.
In even simpler terms, it is being used to describe a way of talking, instead of walking. It's rude, proud or mean manner of speech.
I suggest this because in the New York Times Article, "Deadpool" is being used as a name for a comic book character, and his remarks can certainly be categorized as a sort of swagger, as the word as defined above. You can see this exemplified in The Even-More-Voiced-Over Trailer for the 2016 Deadpool movie currently on The Internet Movie Database, or in the Deadpool series of comic books by Marvel Comics if you have access to those.
As for the word chase it's a metaphor meaning:
- Pursuit with an ardent desire to obtain, as pleasure, profit, fame, etc.; earnest seeking.
Pursuit: The object of one's endeavors or continued exertions or application; that which one systematically engages in or follows as a recreation, occupation, profession, or trade, or with some similar end in view; course of occupation or employment: as, literary pursuits; mercantile pursuits.
The 1914 Century Dictionary and Cyclopedia
In this case, The New York Times is saying that the makers of the Suicide Squadron movie desire to faithfully recreate Deadpool's nihilistic swagger, and tries hard to do it, like a predator tries hard to chase its prey.
Now there are other, more appropriate and esoteric significations of the word Nihilistic, but it probably means exactly what Merriam-Webster said it means in this case, so I doubt it is necessary to explain it much further. I will say that it is being used as a descriptor for the swagger, and more specifically, it is being used as an adjective.