Halo: Reach

*Must be done on Normal or harder difficulty*
This is done on the mission called "Tip of the Spear". You can start at Rally Point Alpha. After you have crossed the bridge with the Warthog, you will hear the AI say that there is a high value target is in the area. An "Elite Zealot". After you go up the stairs and come to a area that you see a turret at the end. Take a left, and this will put you inside of a very large open room. Stay there, and you will see the Elite Zealot come out with a Red Arrow above his head. Kill him quickly and this will be done.

Thanks to Sidedish120 over at x360a for the description.

This was actually surprisingly easy to do (on Normal, your mileage may vary on higher levels). As soon as the red marker is shown over the Zealot's head, shoot him with the DMR. 30 seconds later you should have the achievement. If you just stay stationary near the opening you enter the "valley" from, he just rolls back and forth on the far platform. Easy as cake.

It is on Tip of the Spear, after Rally Point Alpha. After clearing the bridge and going up and around, you will hear that you have a side mission which is to kill a red Zealot Elite. Once you reach an Elite on the turret, the Zealot will be on the other side, near the back. Kill him with whatever you got, wait a couple seconds and BLEEP BLOOP!

Video courtesy of RoosterTeeth

I did this on Legendary, and it was tough, but not ridiculously so. I'd recommend clearing out all the Covenant that you can before you cross the bridge to enter the compound... don't be afraid to use your DMR ammo, the other soldiers will die and leave you some more.
After you clear out all the stragglers, jump back on your Ghost (which you should have at the start of this checkpoint... assuming you grabbed one earlier) and cross the bridge, then maneuver it through the blockades and up the stairs. I used the Ghost from the bridge all the way to the Zealot, blasting every Covenant I could see. Then, when I reached the Zealot, I had plenty of DMR ammo to blast away. Took about 10 headshots, maybe less.

Additionally, you'll find a turret discarded just before the Zealot, you can go back and grab it to clear out the rest of the Grunts. It's kind of a fun level... I imagine it's abusively easy on Normal or lower.

The Solution for People That Suck at FPS Like Me

I figured this solution would need a title to point out that I am in no way good at any Halo... but that doesn't stop me from loving it!

I did this on Heroic. Headshot weapons like the DMR work great. But I suck at FPS so I needed something that took a little less skill.

Before you go around the corner to where you can take shots at the Elite Zealot, there's a Plasma Cannon. You'll know when you're getting close to it because the Covenant will be firing it at you.

Once you take out the Covenant, go to the Plasma Cannon and hold X. Then press RB to rip it off it's stand (if you haven't blown it apart already).

Now just head over to where the Elite Zealot is and unload.

No precision or skill with headshot weapons needed. The barrage from the Plasma Cannon will eventually take him out. It took nearly all 200 rounds for me because I missed quite a bit, but I took the Elite Zealot out on the first try this way.

Good luck and hope this helps. I can make a video of this if need be. Just PM me or comment if you're having troubles.

As an addition to Sonic Sleuth, I also played on Legendary. I chose to drive the warthog right into the bunker and use it to wipe out all the enemies. You just have to drive to the left of the barricades blocking the stairs and drive up and around. Having a partner that is good with the rockets helps but if you clear the way all the way to the Zealot, you can park near the pipes and fire the rockets until he dies!

I completed this achievement solo on Legendary without specifically trying to do so. Here were my steps.

First, get out of your vehicle and use a DMR or needle rifle from sufficient distance to neutralize the turret operator at the base of the mining station, who is just across the bridge. You may also wish to land headshots on the other surrounding unshielded enemies.

Second, drive the Rocket Warthog up to the bridge, clearing out any remaining enemies, so long as the turret has been cleared out. Once all enemies at the base have been cleared, bring the Warthog fully to the other side of the bridge and let your companion out. Once they are on the other side, they will not try to get back in the Warthog.

Third, back the warthog up a bit onto the bridge, get out, and use the rocket turret to shower the upper level with rockets for a while until you're absolutely certain it's clear up there. There are quite a few tough enemies up there, so it's worth being thorough, as it will save you lots of ammo later.

Fourth, fighting your way up the mining facility until you get to a plasma turret. Detach the turret and slowly move forward until the Zealot pops out. DO NOT DROP DOWN. You will know it's the Zealot as a red indicator should highlight him on your HUD for you. He will be on the elevated platform across from you, at about your elevation.

Fifth, use the detached plasma turret to eliminate his shields. If you've inched forward, he may not even move as you hit him with it. Use controlled bursts to conserve ammo and keep the grouping of fire tight. If you have the ammo, and he's not moving around too much, you can finish him with the plasma turret, or you may want to quickly switch to your DMR to finish him.

In any case, this one is none too difficult. Just stay back and use the turret effectively. Sometimes the Zealot will get stuck on the debris of the upper level and will not move for long periods. As long as you keep your distance, he should no problem.

This achievement is ridiculously easy. When you come to the area where the Zealot is, (Red Elite), all you have to do is go around the building to the back, and up the stairs. You will end up right behind the Elite, all set for an assassination and an easy 25G.

Mission 5 – Tip of the Spear

Mission starts at 0:34:28 in the walkthrough video below.

Again, I would highly advise watching the video for this as you can skip a great deal of fighting if you know a few shortcuts. I’ve never speedrun this mission in my life, and after watching RC Master’s video, it was relatively easy to take a few shortcuts. However, there is a rather large shortcut that involves going across the supports to the central structure at one point. DO NOT take this shortcut, unless you plan to replay the mission for the Heretic Leader achievement. Instead, go through the structure, and pick up a needle rifle along the way from the Jackals. At one point, you’ll walk down a set of broken stairs that drop off into a courtyard below. Going onto these steps triggers the Heretic Leader to spawn on the far side, so walk down them, but do not drop off. Back up onto the building again, and shoot across the courtyard at him. Killing him nets the "Your Heresy Will Stay Your Feet" achievement.