If you want the animal companion to be able to gain class levels, he needs to take Leadership and make it his cohort, and you need to assign a level adjustment to it. Even then, its animal HD count toward its total ECL, which significantly limits how many class levels it can gain.

An animal companion gains 2 HD for every 3 levels the Druid gains. A Celestial Tiger is a 'Level -7' companion, but if he has Natural Bond that's only -4.

Assuming you give the tiger itself no LA and only use the +2 for Celestial Creature, the tiger's ECL isn't going to be two lower than the Druid character's level until the Druid is level 16 without Natural Bond, or 20 with it. That means the tiger can start taking class levels when the Druid is level 17 or level 21 depending on whether or not he has Natural Bond, provided his Leadership score is high enough to have a cohort only two levels below him. If he takes Epic Leadership his cohort can be up to his own level, but again the leadership score is going to be the limiting factor.