It'll gain 3 additional HD total, for a total of 7 HD, which puts it in the same region as the other suggested 7th Level or Higher companions in the SRD.
It doesn't take classes, it gains racial hit die (may as well consider them to be the Magical Beast class). You can consider HD to be roughly synonymous with character level, except (in general) the bat won't ever gain HD through gaining EXP.
Some people would argue that, being intelligent, they can take class levels (the Celestial Charger unicorn variant is a level 7 Cleric), but I'd leave that up to the DM's discretion. You'll have to work with your DM to come up with an agreeable LA for your Bat, if that is the case.
Your Bat (as a Magical Beast) will gain the following as it gains HD - d10 HP, BAB equal to it's HD, Good Fort and Reflex saves, and 2+ Int mod skill points. He'll also gain Darkvision 60' and low-light vision.
Every 4th HD, he'll gain a stat point (dumped into his primary), and every 3rd HD he gains a feat. Remember that he has to qualify for feats, just like a PC does.