1 - Face Ghirahim's hordes and make your way to the base of the Sealed Grounds.
Follow Ghirahim into the Gate of Time to the past and exit Hylia's Temple to give pursuit. He'll summon an entire army of monsters to stop you. You'll have to defeat them all in waves to get to the bottom of the grounds. Your toughest foes will be rounds of Bokoblin archers, which you should take out with arrows of your own, packs of Moblins, and a group of Stalfos towards the end.
2 - Defeat Ghirahim in his true form.
Ghirahim's true form is, ironically, easier to take out than his previous incarnations. The first phase is simple: beat him back to the edge of the magical platform and knock him off the edge with a final hit. If he tries to fight back, block his attack with a Shield Bash. When he falls onto the level below, use a Finishing Blow to damage him.
As you send him down to lower and lower levels, he becomes more aggressive. His defensive tactics change as well. He'll try to block attacks from two directions, kind of like a Stalfos. Follow his movements and keep hitting him to bash him down. A couple more rounds of this and he'll plummet to the ground and summon a sword to attack.
This part's not too difficult either. Batter his sword with a few hits and he'll counterattack, a move you can easily parry with a Shield Bash. When he's stunned, give him a stab or two to the jewel on his chest to damage him. Before too long he'll summon a larger sword with which to fight. He'll attack aggressively with this blade, so parry it when you have to, and otherwise attack him and he'll attempt to block with it. You'll chip away at the center of the sword, revealing a magical core that's vulnerable to damage. Keep slashing along the sword's weak point to break it, exposing Ghirahim to damage.
A few rounds of hits will finish him off. Unfortunately the revival ritual continued while you two fought, so a moment later Demise will rise up from the seal in his full form and absorb Ghirahim's power for use as a weapon. Prepare for the toughest fight of any Zelda game thus far. You can save and return to the present to fix up your equipment and heal yourself, so just come back and enter the black portal whenever you're ready.
3 - Destroy Demise once and for all.
Demise fights more aggressively than any other enemy you've seen so far. If you're not very reliable at performing your Shield Bash, you're going to struggle. A lot. You'll have to parry his blows to get in an opening to attack. You can also attack him while he's standing still and blocking. Simply slash him from the opposite direction of his sword. If he's holding it low, hit high, and vice-versa. Usually you'll get in one or two attacks maximum before he recovers, so watch for his next move, which often follows immediately.
After about 25 solid hits, Demise will proceed to his second phase. Lightning will strike the arena, and he'll attack with much greater force and power. Occasionally he'll point his sword at the sky to gather lightning energy to unleash your way. Dodge this attack if he presents it and respond in kind. Point your sword skyward and wait for lightning to strike it, then unleash that lightning at Demise when the opportunity arises. This will stun him briefly and deal some decent damage. When one of you hits the other while your swords are charged, the swords will lose their energy.
Most of his standard sword attacks during this round come in waves, each following in rapid succession. Parry all of them in a row and you'll get a small attack window, but he'll quickly recover and resume his onslaught. Watch out for his charging attack as well. He'll run at you like Ghirahim did, except this attack is much stronger. The most reliable way to damage Demise is to keep zapping him with electricity and moving in for some battle strokes.
Demise will topple to the ground and it'll look like you can kill him with a Finishing Blow, but if you haven't damaged him enough he'll just dodge you and get back to his feet. Continue attacking him for awhile and try again on a second or third opening.
If your Finishing Blow is successful - congratulations! You've just beaten The Legend of Zelda: Skyward Sword!