It adds onto his attacks, which is what makes Volibear such a potent offensive threat if you approach him correctly. Sure he's a tank first and foremost, and you should make sure he does that. But he's one of my favorite jungles, and I've actually found a LOT of success with playing him with a Feral Flare back in S4, and with a Devourer's in the preseason. If he's top, a BotRK achieves the same effect.
Volibear has a very strong synergy with on hit effects, something that his ult makes all the more potent. Building a bit of extra attack speed as well as an additional on hit effect to compliment his ult will let him get in way more damage over the course of a teamfight, or make him an incredibly potent duelist against single targets. More often than not sacrificing a bit of early health for some on hit damage benefits me so much more than just going straight tank, and fits into my personal playstyle more as I prefer to have some damage I can play with even while being a tank.
On another note, Rammus. Mmmmm that passive AD boost.
But yeah, that ult adds onto the damage from his normal attacks, however he'll only be dealing the damage of the normal attack to the target, the ones who get the lightning bounced to them are only taking that bonus damage. But that bonus damage combined with his natural attack speed and an AS boosting item lets him rip through carries like they're tissue paper, and outtrade other tanks entirely.