An edged weapon expert is an enemy with a knife and they usually run at you without hesitation so you need to watch out.
The first time you encounter these is near the end of Chapter 5 - The Edge.
After the bit where you pick up the sniper there is a long combat part. Once you have completed that, carry on until you are next in combat. (This should be inside another ruined buliding) There are a couple of enemies which you kill first, then you want to get to cover where you just killed them. After a few seconds one of these knife runners will charge at you, just back off the cover and he should jump over. As he is jumping over just kill him and providing you are less than 5 metres away you will get the achievement.