First off you must complete the quest 'Old Times' in order to unlock the hippodrome races (possibly even 'Blue Hooligans' as well)
Go through the tutorial and it will show you how to ram and trample enemy riders using the
Once in your first proper race either ram or trample an opponent in an attempt to destroy them.
Now I want to point out that I actually didn't finish off the enemy that unlocked this for me. I weakened him and then he crashed on his own later so I don't think you specifically have to destroy them. I would try to anyway to ensure the achievement pops.
EDIT: You do not have to complete either side quest, but you may have to at least start the quest 'Old Times'. Thanks to Chameleon 1977 for the comment.
Let's keep cheevin' together!