Wraithrock's solution helped point how to know when you are the one making the kill, but how to get those kills needed a bit more detail.
After playing through the whole game (Just 3 races) these are the ways that I know how to take out your enemies.
**For those having trouble with the first race (2 laps): Try to grind someone out into the wall right at the start of the race, then make sure you keep going for grind kills. Tap A just to get enough of a lead to line up your cart against their horses and pin them into that wall. If you don't kill them on the first try, line yourself up again and either whip them to death if they have low health or go in for another grind. You don't have a whole lot of time, but I was able to take out 3 in the first lap and the other 2 in the second. Keep at it!**
Destroying an enemy chariot can be done in multiple ways : Whip attacks
: Using the left and right triggers, try to aim for the enemy chariot's horses for maximum damage. Will take many hits to take out your enemy, so best used when enemy's health is about half or lower. Grinding
: Gain a lead on your enemy and try to pin your chariot against their horses so the enemy has no choice by to ride the walls. This grinds down enemy health bars quickly and is my personal favorite choice in ease and effectiveness.Wall Crash
*: Steer your enemy to drive his chariot straight into a wall. Easiest done during turns and usually done by grinding into them so they have no choice but to take that momentum straight into the wall. This must be a head-on collision with the wall, as it might not work if he is angled too much to the side. This can instantly kill an opponent if done correctly, but is very risky. Not only do you have a chance of crashing yourself but you also can lose a lot of ground during the race if you don't correct your steering immediately after. I recommend pulling back on the left stick just as your opponent crashes to avoid crashing yourself.Obstacles
*: There are piles of debris and statues placed throughout the track, and if an enemy chariot's health is low enough, a quick bump into one can easily take them out. The bigger the obstacle, the more damage they take. This is trickier to do, but it is a way of going about getting your kills.
*Both Wall Crashes and Obstacle Kills can sometimes not give you credit for the kill. Always make sure that your cart has the yellow arrows pointed down onto it in the slow motion camera after a take down to make sure you received credit! (Notice added on Soundgoodizer's request.)
This game is simple by design, as such it follows pretty simple rules. If you are behind an enemy chariot, and it bumps into you, you will take most if not all of the damage, so always make sure you have the lead on your opponent before trying to go in for an attack. Don't be afraid to tap the A button to get that small lead, just don't go into a mad dash because the AI will try to keep up and you might end up rushing the race.
Here is helpful video guide by Maka91 for "Emperor's Glory" and other achievements in the game :