In my opinion, you first have to decide whether it is something natural, or triggered by the character.
If it is something natural, it means all animals tend to evolve and improve over time. I would then suggest to use a variant of the Level Chart found in section 2.2 of the Creatures and Monsters, that would determine advancement, plus possibly a constitution increase as per table in section 2.4. In this case, the creature gains no special ability, but you still have to determine its *natural* abilities (such as tracking by scent for a wolf) and improve them accordingly.
If it is something triggered, I would go with JDale's suggestion : come up with a spell list that does that. I would personally make it work like an alchemy list, in that when you have a spell to improve the bonded creature (let's say, a spell that increases its level, or improves its defenses, or gives it a special ability), you have to cast the spell every day for X days before the effect becomes imbued in the creature. As JDale proposed, other spells on the list could be used to interact with the creature (communicate, heal, borrow senses, sense through, share natural ability...). In this case, the animal becomes an enchanted creature after the first imbuing spell has been completed.