The only way I've really been able to make the Executioner work is with headshots and one of the damage mod barrels. Piercing or Drill ammo helps some too. But most of all; Stasis. That pretty much restricted using the Executioner to the Asari adept and the Volus adept.
Otherwise, the Executioner is just too slow.
Try this: Human Engineer
By far my favorite way to use the Executioner. Overload then shoot, and you will one-shot (body shot) pretty much any non-possessed mook in the game (including Phantoms) on Gold. The Power Amp IV is overkill on most enemies, but if you're playing against Geth or Unknown enemy, you will want the extra Overload damage because you won't get the benefit of the extra Neural Shock damage.
Playstyle for most enemies: Overload - Shoot (dead) - Overload - Shoot (dead) - repeat. You can usually use Overload as a reload cancel, because this build keeps cooldowns really low. Once you get the timing down and get used to switching enemies quickly, you can mow down mooks rapidly at any range.
For Dragoons, Pyros, Etc., you may want to throw an Incinerate in there. For Dragoons, I start with Incinerate, fire once, and then Overload for a Fire Explosion and dead Dragoon.
For Bosses: Alternate Overload/Incinerate for Fire Explosions and Tech Bursts, with one shot between powers, using each power as a reload cancel. Puts out an impressive amount of Boss damage.
EDIT: For Platinum, you may want to try to squeeze a little more health damage out of the Executioner to ensure one-shot body shots. Warp or AP Ammo might be better there, along with the Pistol Amp V.