This achievement is a bit annoying but, I found an easy method that can keep you on the safe side.
First things first, you need to know what you're going to be doing. After visiting your mom for the first time and completing the mission that comes along with it, the L.C. Cage Fights should unlock. There will be 5 rounds and then a final round in which you'll be named a champion. Each round has 3 people you must fight. You have to do this in one sit down and if you die you'll have to start all over again. In other words, you must survive to be the champ.
Here is where my help comes really handy.
After every round you fight, you regain very little health back and you get an option saying "continue fighting" or lose 50 bucks and "spectate." SPECTATE. When you spectate just gamble on whoever you think will win and then continue fighting until you make it to the final round. Doing this pretty much gives you all of your life back and ensures that you have a better chance at staying alive on the future fights you still have to survive.
During the third round, people will start coming out with weapons. This is where things get tough and at times, tricky. If you want to survive easier try to take the weapon from their hand by countering (by pressing back A then forward B) when they attempt to attack you. The weapon you take will stay with you until the end of the round so it's a great ally if you want smooth rounds. If you're having trouble with counters, no worries. You can also try getting them in a corner and making sure they have trouble moving. If you knock them out and you didn't take the weapon out of their hands, you can't pick up their weapon. It just disappears for some strange reason so try taking weapons frequently after round three.
Hopefully you won't have too much trouble with this. Took me a few tries until I found out my little spectate trick. Good luck!
EDIT: Before the server crash I posted up a video that RoosterTeeth made to help those who are better at visual instructions rather than a written guide. I've placed it once again for those who need it or would like a better understanding of it. Hope it helps.