The one handed axes are the weapons in the second row of the primary weapons in the Man-At-Arms class. To get this achievement you will need to get 25 kills with the first one handed axe weapon that is unlocked.
I personally haven't noticed any preference in game mode for these weapons as they are a good all rounder. To make this a bit easier you should accompany your weapon with a shield as it makes blocking easier.
Any suggestions for this solution just comment. :D