Fallout Shelter

In order to do any quests, you first need to build an Overseer's Office, but first you'll need to unlock the Overseer's Office by having 18 dwellers in your Vault. The Overseer's Office requires 2 rooms' worth of space.

Soon as you build the Office, you can begin sending out your dwellers on quests. Each quest requires one team of up to three dwellers, restricted only by certain levels (Example: if it says "Level 22" on the bottom of the quest card, then you can only send out dwellers Level 22 and up; and occasionally you will be limited as far as weapons, for example one mission I did required my team to only use pistols.). You can provide your team with supplies of stimpaks and radaways (Always recommended) before sending them out. If you have any rare/legendary characters and/or weapons (and outfits), I recommend using them on your quests to easily clean house with enemies. Certain pets also reduce the amount of damage their equipped Dwellers take, so if you have them, use them.

There is also a time stamped on the quest, showing exactly how long it will take your team to reach the quest's location. You CAN speed up the travel time using a required number of Nuka Cola Quantums, but unless you have an excess amount of NCQs (or real-life money), you can simply send the team out, save your game and turn it off (optional), and go do something else (like watch a movie, or go run errands, etc).

A few times during quests, you can finish the objective(s) before you finish exploring the location of the quest. I prefer to fully explore the location for loot (weapons, junk, outfits, NCQs, and theme recipes), which also leads to more enemy encounters. Unless your team is extremely low on health and (more importantly) supplies, keep exploring the location even after the quest is complete and you need to return to your Vault to turn the quest in. Once you've covered every section (except for elevators since they're small rooms and have nothing of value hidden in them), then you can head back to your Vault.

Keep doing quests and exploring every room in that quest's location, and eventually you will get your 500 enemy kills.

Some have suggested altering the time on your PC or Xbox One to speed this achievement up, but I personally do not recommend it unless you're in that much of a hurry. If you get it through that method, more power to you.

UPDATE (4/5/17): For those who seem to be stuck at a certain percentage towards this game, I recommend creating a new Vault (you can make up to three separate Vaults) and starting over. Yes, it's tedious, but so far it's the only way I know of that will overcome the "percentage glitch" that plagues some of the game's achievements. I had to create a new Vault when my progress for "Get Off My Lawn" stopped at 86%, and I managed to unlock that achievement using that method. However, I had to do all 50 Raider attacks again to get it. To check your progress, go into Stats (looks like a wallet in the Pipboy menu).

Have fun!

As soon as you have 18 or more dwellers in your vault you will unlock the Overseer's Office, you will use it to send a person or group to a quest, where at the end of it you will have some rewards, doing the quests you will encounter several different enemies, just kill as much as you can and complete the mission, you will also unlock some achievements on quests, for exemple: