Clint had been read in on certain things about Red Skull, most SHIELD agents were at least aware of the HYDRA leader and his insanity, not to mention all his mad schemes. HYDRA were still an issue, just in a different way than they used to be.
It made sense to Clint, Steve being the only known success story of the original plan for the Super Soldier Serum. Clint understood that Fury had methods and he had reasons for keeping things close to the vest, but sometimes it didn't seem logical.
And really, Clint wasn't breaking any serious rules. Cap was his team leader, his team leader needed to know this stuff. Ergo, Clint was just being a good subordinate for once. "Well, we're gonna need an inside computer, so, you up for a small trip?" They didn't need to go as far as the Helicarrier, since Clint's presence would likely mean Fury would be informed the moment he started using one of the access points, but the Headquarters in New York gave just as much access with a little less security.