Chivalry 2 (UP)

If this is your only achievement you are going for the fastest way to get it would be to join into multiplayer games and make sure you are spawned in as the archer class the fastest archer class for this is unlocked immediately and is called the longbowman, pick that one. Often-times on your side of the map by your spawn there will be these chests that are half open and filled with arrows called ammo crates.

Simply find one of these and if no one is nearby from the enemy team you can just repeatedly tap which will fire an arrow for you in the fastest way and repeat until you are all out of arrows. Then go to that ressuply crate and hold on it to refill your ammo. Repeat this until you have all 1,000 arrows fired.

Update: I'm not sure if firing the arrow is all that you need to do as some people are reporting that you must now HIT the arrows.

If this is the case the bow and standing near a resupply is still your best option as having ammo and being able to fire fast are still highly important when it comes to getting this quickly. Make sure you adjust for drop as arrows do drop the further you try to fire them.

Update 2: It is also possible this is no longer counting for bots as reported by one person but I am not sure. These also needs to be arrows from your bow and not your crossbow.

The absolute fastest way to do this if you're strictly trying to knock this out is by going into an offline practice match and set it to free for all.
Map: Tournament Grounds
Score limit: 50
Time Limit: 14
Number of bots: I always put 32 so they can all keep each other busy

Now when you spawn just run into the middle-ish part of the map in front of the stairs that has the throne. Not on top, on the ground standing on the wooden platform and just spam arrows at your feet the entire time. You'll have unlimited ammo by picking them up and the bots for the most part should completely leave you alone.