The Chase of Captain America was an attempt to capture Captain America and prevent him from discovering the truth behind the assassination attempt of Nick Fury and the continuous existence of HYDRA, the supposed long-defeated terrorist organization, embedded within S.H.I.E.L.D.'s ranks.

Background

S.H.I.E.L.D. Director Nick Fury was attacked by HYDRA operatives, disguised as Washington, D.C. police officers, and the Winter Soldier in an attempt to kill him.

Before Fury's apparent death at the hands of the Winter Soldier, he gave Rogers a USB Drive, obtained by Black Widow at the retaking of the Lemurian Star, which Rogers hid in a vending machine at Fury's hospital. 

Alexander Pierce questioned Rogers in order to get information on the matter, but was unsuccessful.