To unlock a weapon execution you have to get 25 kills with the following weapons (default weapons):
Lancer Rifle
Hammerburst
Retro Lancer
Gnasher Shotgun
Sawed-off Shotgun
And 10 kills with the following 'pick up' weapons:
Gorgon Pistol
Boltok Pistol
Boomshot
Digger Launcher
Hammer of Dawn
Mulcher
Mortar
The Oneshot
Scorcher Flamethrower
Torque Bow
Longshot Sniper Rifle