The Bulldog machine gun is the gun you start out the game with. Accumulate 50 kills with this gun.
Note: Gun kills apparently stack even after death or loading from a checkpoint. If you find a good point in which you are killing many enemies with this gun and you've just saved, press start and select "Load Last Checkpoint" (or whatever it is). You'll reload from the last checkpoint but the game's internal counter doesn't. Thus, the amount of kills you had with this gun prior to your reload will not return to the point of that save.