Halo: Reach

I just want to point out there is no foolproof way to earn this, just a bit of strategy and luck.

The only slightly tactical way to get this is during a race/rally matchmaking session. In most cases most other players will be too preoccupied trying to hit those checkpoints to care about fighting meaning if you hop off your Mongoose (Hold X) you just need to kill someone speeding around. The other thing to note is jump out of thier way if they are coming at you, don't let yourself be the 1st strikee.
The other half tactical option is to play Team SWAT, As you have no overshield and a headshot is instant death it is more a bit of luck at just being the first shot off, something which you can do in a small map is lob a grenade over to where the other team spawns and cross your fingers.

Short of this it is just a bit of luck that you spawn near enough to a player who you manage to kill before they do you.

To get the Achievement, you need to get a "First Strike" Medal, which means get the first kill in a match. (Please note it only works in Matchmaking, NOT THE FIREFIGHT GAMETYPE!)

I got it when playing Team SWAT, no shields so Headshots are instant kills/ So just select the Team SWAT PLAYLIST and hope for the best.
If you don`t want to do it in SWAT, just continue playing any Gametype and try to get the Power Weapons (Sniper, Rocket, Plasma Launcher etc) and try to lure an enemy towards you while you have a Power Weapon.

EDIT----

This is now much easier done in the Living Dead Playlist (Zombies), there are 3 rounds in which you can earn 3 First Strikes in total, so a pretty high chance to get it. My tips are when you spawn as a Survivor equip your Magnum and try to headshot the first Zombie in sight, you will headshot him with one shot because shields are disabled for the Zombies, and if they get close, shotgun them back to where they came from! (Best maps to get Headshots on are Powerhouse, Countdown, Boardwalk and The Cage.

Thats all I can do, good luck.

This requires you to make the first kill in matchmaking game. It doesn't have to be Slayer, the game can be objective based like Oddball or Assault, and team based or lone wolf style.
A good idea of initial spawn locations may aid you in trying to obtain this, so you can rush the enemy spawn to get a quick kill. Also, on a more open map (such as Haemorrhage), perhaps picking the sniper rifle would allow a quick kill at range, or jumping in a Ghost or Revenant may also work.

I got this one on the third round of a CTF game. Each round counts for a new game so you have the start of each round to get it.

Got mine in Grifball. All the rounds last like 30 seconds to a minute. Its 3 on 3 and everyone will come at each other with gravity hammers so just swing early and keep trying and you will get it in a few minutes.

Easiest in SWAT, since roughly half the games you play in Team Slayer and BTB will end up being SWAT. Just sprint ahead of your team at the beginning of a round and try to nail a headshot on the first person you see.

Edit: as of the October update SWAT has been removed from TS and BTB and now has its own playlist (thank sparky).

Easiest to earn in a game of SWAT. In Swat, you have no Overshield, only health bars, so one shot with a DMR to the head is an insta-kill.