To unlock "Executioner" you need to perform 100 executions.
This can be either be in single player, online multipler (co-operative) or online multiplayer (solo).
Any execution counts, including enviromental and dual executions on any map.
Chances are, if you have this game, you can pick this up without any hassle or need for tips.
a) If you've not finished the campaign, or aren't level 100 yet, I'd suggest doing them purely in those respective areas. Two birds with one stone.
b) If your going to do it in the arenas, pick Jupiter as your God. Focus makes this simpler, easier, and faster.
c) Focus gain gear is your friend.
It seems double executions only count as a single execution, even though a post match screen in arena will show as two executions.
In solo arena, to test, the counter worked perfectly.