The Battle
To begin with, you should re-enter and exit the courtyard. Why? Well, this way you will begin with Griffon on the ground in front of you, instead of up in the air. This won't make a huge amount of difference on Normal mode, but will give you a decent head start on Hard/DMD mode.
Griffon 1 can be fought both long range with guns and with short-range melee attacks, though the latter works much better if you have Ifrit abilities such as Inferno and Rolling Blaze. Incidentally, if you have red orbs to burn, you can run all the way back to the divinity statue in the pillars area - Griffon will still be waiting for you upon your return to the field.
To fight Griffon at long-range is pretty straightforward - you can plug away with either E&I or the grenade gun. I favour the grenade gun personally, but E&I does have one advantage which the GG doesn't - you can make Griffon fall down from the sky with E&I. It takes a while, but if you plug away for long enough, eventually he will drop down from the sky like a stone, where he will lie on the ground and be vulnerable for a long time. Plug away like mad with DTed E&I and/or charged DTed Roundtrips during this time.
You can't make Griffon drop down from the sky with the GG (actually, technically you can, but you can't do so with the GG alone), but the shots do more damage than E&I. For such a big creature, Griffon is surprisingly easy to miss - hold R1 and make sure that the gun is pointing straight at Griffon. If it isn't, then keep jumping backwards until it is.
Grenade-rolling - cancelling the lag of a grenade shot with a roll - is a helpful technique to know for this fight, as is DT-cancelling. If you fire a grenade at Griffon in DT mode, then quickly de-DT before the grenade hits Griffon, you'll get 2 runes of DT back upon impact. Not too shabby. Being in DT mode as much as possible is recommended, because a) DTed grenades do more damage and b) you can cover more ground at a faster rate, which is useful.
Incidentally, don't use Air Raid - at the very least, don't use it when Griffon is up in the air. If you do so, you'll incur a *horrible* camera angle change, whereby the camera zooms in upon Dante and Griffon disappears from view - his counterattack attack become impossible to avoid when you're floating around in mid-air. Even on the ground, using Air Raid is a bit dodgy, but still better than when Griffon is up in the air.
To fight Griffon at short-range with Ifrit, you'll need Inferno and Rolling Blaze - and preferably Meteor Lvl 2. You can knock down Griffon on the ground in a similar way to knocking him out of the sky. It doesn't even take that much - 4/5 Infernoes, combined with a few Rolling-Blazed jumps through Griffon, will topple him over. Try to preserve some DT for this part, as, at such close range, you can do enormous damage. Either fully charge up a PPKK combo or as many Meteor 2s as your DT will allow you to.
It's much harder to knock down Griffon on the ground with Alastor - hacking away at his legs for long enough will do it, but it's a lengthy process and probably not worth the effort. If you have Stinger Lvl 2, then feel free to use Stingers against Griffon whenever he hops in close to you.
If you possess Roundtrip/Meteor Lvl 2, then you can attack Griffon 1 at long range on the ground. You can safely charge up either attack immediately after any one of Griffon's attacks has finished.
Griffon's lightning attacks on the ground shouldn't give you too much trouble, but if they do, there's a couple of ways you can bring them to an early end. One way is to use Vortex Lvl 2 - simply burrow your way towards Griffon to stop the attack in its tracks. Another way is to get very close to Griffon - once you get close enough, Griffon will do 1-2 "wind-up" V-bolts and finish the attack - the V-bolts are usually easy enough to jump away from or roll under.
No matter what you do, you can't stop Griffon rising into the air after a certain amount of time has passed. On Normal mode, this is no problem really, but on Hard/DMD mode, his chained web of lightning attacks can be hair-raising. I've found that if you refrain from attacking Griffon at all when he is up in the air, he will usually come down a lot sooner than if you keep on plugging away with grenades - however, being the l33t player I am, I just keep on attacking :p
If you've done a certain amount of melee damage on the ground, then E&I should knock Griffon down from the sky quite quickly - even the GG will bring him down if you brought him very close to the point of being knocked over before he rose into the air.
Finally, always make sure to get the last hit of the battle from a distance, as Griffon will surround himself with a shield of electricity bolts upon defeat which will hurt you if you're too close.
One last note: ignore the pink discs which Griffon lays about the battlefield on a regular basis. You can jump on them, and that's about it - I've yet to discover any useful purpose for these discs.
