An Edged Weapon Expert, or Bayonet Runner, is a special type of enemy that charges straight at the player with a bayonet in an attempt to melee you. While, if they do get close enough to melee you, they will kill you instantly, and they are even capable of dodging gunfire. With that said, these guys never take cover, they never use a gun (even though they are equipped with one), and have very little armor, so they can be killed very quickly if you are quick enough to spot them.

They will come at you from afar, so in order to get this trophy, you will need to kill them while they are, at the very least, 5 meters up to you. Your teammates call out special enemy types if they are present, including these guys, so listen to what they call out in gunfights. There are many times you can get this trophy in your 2 playthroughs, but there is a great spot for getting it. 

The very first time Edged Weapon Experts are introduced are in Chapter 5 of the story, "The Edge", after the roof under you collapses and you drop down. This is also the earliest and (one of the) best place to get this trophy. Once you drop down from the beam, retreat and take cover in one of the pillars at the back of the area, then keep a lookout. You should see the runner running right through the gunfight with his bayonet. Once you see his, aim at him but don't open fire until he is close to you. Once he is, kill him to get the trophy.