This achievement is bugged, and there is a huge chance you won't get it either.
After 16 years we can't really hope for a fix but there is something worked for me, but i'd suggest you to complete achievements first you have progressed so far, just in case.
You see, when the game starts at the first time, all achievement progress are 0, BUT the first death restores previous progress. We need to run the game for the "first time" again, for that we need to get rid of saves.

-First Open the game folder from your Steam library, it should be at (your drive) > Steam > Steamapps > common > Source SDK base 2007 and leave it open
-Remove the game using Steam library.
-Navigate back to folder and remove (or move to somewhere else if you want) everything associated with Age of Chivalry.
-Reinstall the game, and start it.
-Before first death, get the 10 kills with hand axe, by using a friend or bots, and trigger the achievement once more.

Hopefully everybody will be able to get it this way, good luck!