When Doing the security missions on the first island, you will have to build an ACU center. When it's built, press A on it and select the helicopter and take control of it. Fly the helicopter close to your dinosaurs and press Y to pull out the gun. I found the best way to make a dinosaur run is to shoot it with a tranquilizer. Small dinosaurs go down in one hit, so shoot something big like a Triceratops.
When I got the achievement, the Triceratops ran after the 2nd shot and my third shot hit it as it started to run.
I do not know if the achievement pops if you hit a running dinosaur that is not fully tranquilized. Please comment if you know for sure.
When the dinosaur goes to sleep, you get this achievement:
Hit a dinosaur with a tranquilizer from 125+ meters for this achievement:
While you are at it, shoot a dinosaur in the head for this achievement: