Sniper Elite V2

By playing through the campaign, you will likely already have a number of these. I myself had 58/100 after completing the game.

A good spot (IMO) for farming any remaining kills needed is on the second (first after the prologue) level, Schöneberg Convoy.

Get to the room you remember from the demo (which you get to after killing the first three enemies and has a enemy with his back turned to you in it, with the German leaning against the doorway down the street). Move downstairs and position yourself against the window directly to the right of the exit door (if you were looking at it from outside, left from your perspective). There will be a number of Germans down the street. Fire off a shot, and they will scramble for cover.

On the right hand side at the end of the street there will be a lamppost with a car just behind a pile of rubble, with what looks like a wooden dresser to the right of the car. Germans will take cover behind this dresser, then will move out and start running down the street towards you, giving you an easy moving kill. Once you get better at leading enemies, you can get more from the Germans running across the street perpendicular to you.

Once you kill all but one or two Germans two more will spawn directly beside the car and take cover in the same spots. I found I could get 4-6 moving kills every three minutes or so. Just keep reloading the checkpoint after you kill all the Germans and repeat until you have 100 moving kills.

Be aware there is a sniper at the far end of the street and an MG42 in the building directly across from you. Obviously this is easiest on Rookie mode. Additionally, there is an explosive barrel at the end of the road, so feel free to shoot it before reloading to work towards Detonator.

Finally, this game does not track the kills in a logical fashion - some kills tagged 'Kill' (because you did not get a killcam shot on a moving target) do in fact count. I know this as I was keeping track and got 75/100 when I only had 69 on the tally. So just keep at it. If you really need to know, just quit out, check your stats and continue from the checkpoint.

Good Luck!

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Load level 1 prologue on cadet. Disable the tutorial if you have already beat the game. As soon as the level starts (skip the dialog) shoot the russin handler on your left. Then move over to the target Von Heisenberg. He will look around, draw his gun and take cover behind a broke pillar. He then stands up, turns and start running (shoot). He runs directly away from your position making the kill almost like a standing target. Every kill I got the target was on the left side of the pillar. If you miss you will have plenty of time to kill him as he runs behind the pillar, moving right towards the jeep. The kill must say "moving target" for it to count. After the kill is scored restart level. Took me about 20 second between each kill.

Note: "moving headshot", "moving kill" and "moving vital hit" did not count towards this acheivement. Check your status at the main menu under extra/career stats/scoped moving target kills

I hope this helps.

The solutions already provided are really good, but as an alternative I used the following method so I thought I'd share it in case it's helpful...

I found mission 8 (Karlshorst Command Post) to be a great place to grind out this achievement. It's over halfway through the mission so if you're going for a stealthy approach and/or you're not quite sure where you're going then it might take a bit of time to get there. However, if you're just working through the campaign then this is not an issue as the location is unmissable, and once you get to the "perfect spot" it's a really quick method to get the achievement.

Having said all that, Har Miggido pointed out in the comments, if you know where you're going, or if you've already done the campaign then after loading the level: "You can leg it to the tank, destroy it and take out the guards at entrance (where the next checkpoint is) within 3 minutes. No need for stealth."

After you've destroyed the tank, taken out the guards defending the command post, and entered the building you will get an autosave. When you go down the ramp there is a long corridor along which waves of Russian soldiers come running towards you. If they spot you they stop running (or begin walking which means it won't count) and start shooting at you. So the trick is to stay at the end of the corridor (at the bottom of the ramp) and strafe left and right, either side of a pillar in the middle. As soon as you see someone running take a shot. If you miss and they stop and start firing then move to the other side (which moves you out of their sight) and they normally start running towards you again.

It seems that the first wave are the bravest and will run towards you. After you've taken them down you get a load of less aggressive soldiers who hang back, but you can still catch them running (normally back to cover).

You can keep shooting them until the music stops as this indicates that the only person left is the officer with the intel. Do not kill him otherwise you will trigger the next checkpoint! However, he will helpfully stay out of sight around the corner at the end of the long corridor so there is no risk of killing him by accident. At this stage you can reload the last checkpoint which will start you at the top of the ramp and you will be gifted with more waves of suicidal maniacs who are hell bent on giving you this achievement as quickly as possible. I got pretty good at sniping them whilst moving and I was getting 5-10 moving kills every 3 minutes. By this point in the game you might already have about 50 "scoped kills of moving targets" so it shouldn't take more than 15-20 minutes to finish it off.

I hope this helps.

Another solution which also gives you the Potato Masher and the Detanator achievements:

Mission 6: Tiergarten FLAK tower. When you enter the actual tower, there is a checkpoint.
Run up the stairs from left or right. Throw a grenade in between the two Germans. For some weird reason, the game registers the dodging soldiers as 'running away'. It also lets the two gas tanks explode, and adds kills by explosion. 3 achievements off the list! :)
Just keep reloading the checkpoint until you get the achievement!