To unlock Explosive Resolutions you must kill 1000 zombies with explosives. This challenge started at midnight GMT on 30th December 2013 and runs for 31 days.
This will take you anywhere from ten minutes, to ten seconds, depend on what category you fall into it.
The reward is a shiny 0g challenge, and an ingame hat (New Years Hat).
Group A - People Who've Completed The Game/All Blueprints
For you guys, this is a ten second job, if you were thorough throughout your playthrough.
Load chapter 8, and go to the nearest safehouse. Go to the weapons locker, and get a "Massive Bomb". The blueprint for this is unlocked by beating the game (Overtime included) if you have collected the pre-req blueprints. If not, two of the lesser bombs should suffice.
NOTE - If you do not have a Massive Bomb in your locker, for it to appear here, you MUST have made it yourself once manually. Go to your blueprints page, and craft one, using the locker to spawn the components if needed.
Make two, just to be safe. Now, look around, find a massive group, throw the bomb into the middle and wait for the nuke. As chapter 8 is filled with zombies, this shouldn't be hard. I did it right outside the first garage/safe house you go too. I got 1057 kills with one bomb. If you're short, don't worry, you've got two. Toss a second into another group and then enjoy your new hat.
Edit - BlazeVII has pointed out that the X-Busters charged shot also counts as an explosive, which would be a great way to mop up after using a Massive Bomb.
Group B - People Who Haven't Finished The Game
This could take a bit longer, depending on where you are at and what blueprints you have.
Load the latest chapter you have (there is usually more zombies) and go to a safehouse (chapter 4 starts at a safehouse).
Now, you want to build the best explosives you can.
Look out for any of the BOMB prints (Huge Bomb, Enourmous Bomb) etc, as they're pretty sweet. You can make two of these.
Failing that, you can swamp your inventory with the combo explosives (you can make 4). Each of these gives you 9 ammo (so 36 bombs).
If your using the smaller explosives, I'd suggest either I.E.Ds or Dynameats. However, the speed should all be the same. Bomb Toys and Jack in the Boxes also work well as they attract the Zombies like Dynameats. Explosive Chairs, Helmet Bombs, Sticky Bombs and some of the bears with explosive weapons will all work.
Once your inventory is emptied, if you haven't reached 1000 yet, just reload the chapter and start again. It shouldn't take too long.
Happy hunting and Happy New Year!