No body no death. That scene cuts to a wide shot of a muzzle flash and thats it. Also that trailer made it seem like at least the kingpin scene was a flashback to well before the events of hawkeye.
He shows supernatural strength and toughness in the Hawkeye series. He shrugged off a bomb to the face and a car and an arrow to the shoulder. He probably got something like the supersoldier serum in him now.