You will need to do this on two different battlegrounds: the Everglades and Mexico.
The Alligator: You will be able to get this in your first match of the second chapter of the campaign. You will notice an alligator on both the left and right side of the ring.
To feed your opponent to them, hold
to carry your opponent and then walk towards the centre of the ropes, aim directly to the left and press
to toss them into the gator. Easy to do, brutal to watch.
The Ram: You will first get the chance to ram your opponent in the campaign (see ya later, alligators).
In Mexico during the third chapter, just after the Everglades (Florida), you will face off against Daniel Bryan with TJ Salazar.
To control the ram, exit the ring to the top and head to the top left corner to interact
with the glowing controller. You will now control the ram for a short period of time.
to jump into the ring and
to attack, you can also just run into the opponent and it will automatically attack them. Just chase D Bry down and hit him to unlock the achievement.
If your opponent uses it first, or you fail to ram the opponent, you can try again a bit later in the match. It has a cooldown period between uses but it doesn't take too long.