After completing all 62 other substories, enter New Serena as Kiryu to get a letter. A car is now waiting for you in the underground parking lot. Grab a full load of weapons, healing items, and accessories, then go to meet the car as Kiryu.
This is the fight you've been grinding for throughout most of the game, and doing it on any difficulty higher than Easy is just masochistic. If you're going after the Indomitable trophy, clear the main game first, then go after Amon in Premium Adventure mode and dial down the difficulty once you start having problems.
Akiyama's up first, fighting Kazuya Amon. He's ridiculously fast, and if you just go after him with your standard kicks, he'll dodge and encounter so easily that it may actually make you feel kind of bad as a person. At very low health, he goes into HEAT mode and becomes almost impossible to hit with standard attacks, which is a good time to shotgun Tauriner/Staminan drinks and catch him with HEAT actions against the arena wall.
Saejima is up next against Jido Amon, who's armed with a sledgehammer. This may be the most difficult fight in the game, as Jido's hits don't inflict less than about 15% damage, and once he reaches HEAT mode, he can actually combo you against the wall for pretty much your entire life bar. He also instantly counterattacks any attempt at a grab while he's in HEAT mode, which sends Saejima ragdollin' across the arena.
Fortunately, Jido doesn't stay in HEAT mode for very long periods of time, which lets you drop him on his head repeatedly as long as you pick your moment carefully. What makes things difficult is that Jido actually regenerates life while he's in HEAT mode, so this fight can last more or less forever unless you stay close to him... and if you stay close to him, he may just decide to cave your face in on a whim.
It is absolutely pointless to go into this fight with anything else than an absolutely maxed-out Saejima, carrying nothing but Staminan Sparks, wearing the best armor you can dig up, and with every HEAT move and minor stat buff you can wring out of the game. Run the IFR-7 until your eyes bleed. The basic idea is to do whatever's necessary to string together as many HEAT moves in a row on Jido as you can, using items, training, and sheer practice to counter his ridiculous health regeneration.
By comparison, Tanimura's fight with Amon Sango is almost quaint. He's kind of like Nair with a machine gun.
Wear something like the SWAT Body Armor or the Sacred Tree set to protect against his weapons fire, then do your best to stay on top of him. Sacred Tree in particular utterly defangs him, as he can't do much to you until he reaches HEAT mode. At that point, he'll try to catch you with a very similar parry to the one Tanimura uses, so don't attack blindly; watch for it, wait it out, and counter his counter.
Finally, Kiryu gets to deal with Amon Jo, who's a one-man boss rush. Early in the fight, he uses explosive throwing knives, but once he hits HEAT status, he starts using the other Amons' weaponry. When he's got a hammer, he regenerates like Jido; when he has a gun, he'll fight like Sango did; and when he's bare-handed, he uses kick attacks like Kazuya did. He can also call in an airstrike on the arena, which can hit you for an immense amount of damage if you're standing inside its signal flare when it hits. Don't bother trying to attack Jo when he's in Jido "mode"; he's going to regenerate a bit and there's not much you can do about it. In a weird way, he's most vulnerable as Sango.
When Amon Jo drops, you receive the Amon Defeated trophy and the Divine Dragon Short Sword. You can also pick up Amon's Sunglasses later on at Naomi's Palace.