How long does it take to get your Chocobo to level 10

[ChameleonMS]

If your chocobo is stabled, you can feed each hour. Another member of your FC can also feed it each hour. Each feed grants 1% XP (assuming a clean stable). If the stable is not clean the XP granted is reduced.
As has already been mentioned, challenge log is great source of XP for bird.
If you use your bird in a fate, it only gains XP for the kills, no XP for the fate completion bonus.
If you kill in the open world, you need to be killing a mob near your level.

It has been a while I remember joining Spiritbonding parties and we would massacre the mobs, bond our gear, and level out chickens all at the same time. Spiritbonding is good source for DPS of getting V and VI battle materia. Just buy NQ from the vendors, put cheap junk materia in it and make a nice profit while levelling that bird.

All that said, you are still looking at a level couple of weeks if you dedicated.