That video that is linked is advice on leveling a chocobo, but the person in the video isn't even max rank on his bird.
My name is Project Stfu on Sargatanas server. You can contact me with questions in game about it if you want.
This is me: http://i.imgur.com/0IdNTjU.jpg
There are a few ways to do it, but I am going to tell you the single fastest way I know of to level your chocobo and what got me to rank 10.
It is still not something you can do in an afternoon, it will take a few sessions and probably a week to get max rank.
First go to the following spot I marked on this map: http://i.imgur.com/7xj7rjS.jpg
Once there you will notice a ton of level 50 monsters in the area, now you grind them. Preferably you are on whm so you can stoneskin yourself, collect them all up and Holy them dead, repeat.
Now solo this is slow, so to maximize your speed, you will want to invite 3 other people and make a full party of 4 people + 4 chocobos.
There is a ice rock in the middle of the cave, when you gather them use it to LOS them all so they group up, then AOE them down.
This will grind you several hundred thousand exp per hour.
That is all there is to it.