Marvel's New Bullseye is Actually [SPOILER] in Disguise
This article contains spoilers for Hawkeye: Freefall #6.
Marvel's new Bullseye is actually the Avenger - Hawkeye. Bullseye is one of Marvel's most dangerous killers, a ruthless assassin who never misses. Ironically that makes Bullseye easy to beat by Daredevil, who can predict every shot will be a kill-shot, and thus deflect them with apparent ease.
Daredevil isn't the only hero who has a history with Bullseye. The assassin has frequently crossed paths with Hawkeye, the Avengers' own master marksman. In fact, when Norman Osborn suited up supervillains as his Dark Avengers, Bullseye got to dress as Hawkeye for a while - albeit a far more dangerous version. In Matthew Rosenberg's Hawkeye: Freefall miniseres, Bullseye decided to repeat the idea and steal Clint Barton's current Ronin costume. That even gave him the chance to put Captain America in hospital, because Steve Rogers had no idea he wasn't talking to his fellow Avenger.
Fortunately, in Hawkeye: Freefall #6, Clint knows exactly what to do to put matters right. Bullseye had left his own costume behind when he switched outfits, and Clint figures it's time to become Bullseye for the day. That will give him plenty of plausible deniability, because let's face it, nobody is going to associate a murderous assassin with an Avenger.
Hawkeye's arrival is well-timed; Bullseye has headed to the New York train station, intending to see what kind of mess he can make if he crashes two trains together. Naturally, there's only a brief conversation before the arrows start flying; both Bullseye and Hawkeye are out for blood. Each uses their own arrows to intercept the other - and then, to Bullseye's surprise, one of Clint's gets through. Having proven himself the superior marksman, Clint Barton ends the fight in classic fashion. His next arrow is a trick explosive one, and the real Bullseye is successfully taken down. What follows is brutal justice on Clint's part, as he sends a steady stream of arrows into Bullseye, penetrating his hands and his calves. This is a beating the real Bullseye will remember for a long time - and he was given it by a superior marksman dressed in his own costume.
Bullseye may be Marvel's top assassin-for-hire, but he has a habit of trying to punch above his weight. In this case, his attack on Hawkeye became deadly personal, and prompted Clint Barton to go to extreme lengths to take him out. Presumably the Bullseye costume is simply dumped in a trash can afterwards.