Dino-Rama is the third event in the Animals & Friends booth of free play. To unlock it you'll need to play through the first two events there, or you'll eventually get through through Adventure Mode.
Dino-Rama is a shooting game. Concentrate on shooting the pink sea monsters at the bottom of the screen. After shooting a certain amount, a hydra (multi-headed sea monster) will appear in the center of the screen. Shoot that down, and then go back to shooting the single-headed sea monsters. Do this eight times and the achievement will pop.
You should have plenty of time at this point to shoot other targets on the screen to go for goals.