I remember this one. He's a hard one, and I hated fighting against him.
Here's what I did.
Some say that beating the Tentacles doesn't really do any good, since it only stuns the enemy for a short period of time and then they rejuvenate.
Personally, I say it's the best strategy. Not only does it keep you away from the middle, where the poison from the enemy continously leeches away at your HP, but it also means that the Tentacles will attack you, bringing down your HP steadily but slowly.
This is what I did.
Stock up on not only Curas, but also Hi-potions, Elixirs, Potions, whichever works best for you. Don't rely on just magic cures because those can take a while and one good whack could completely delete your HP.
Now, go around and stick it to those Tentacles. If they go to whack you, let them.
The point of this is to get your HP down till you get to a Limit Break, and then unleash the Limit Break once the Leechgrave is stunned. It knocks out about two to two and a half HP bars alone before the stun is over. Heal yourself, and then repeat.
It's much easier than not going for the Tentacles, taking twice as long and possibly dying along the way.