In this chapter, your goal is fairly simple: hunt down Dark Pit. The interesting thing though is that Dark Pit is not a part of Medusa's Army; he's his own faction. Therefore, he and the Underworld forces will fight each other as well as you. Use this to your advantage by letting them take each other out when possible.
The chapter begins with Dark Pit attacking you in a nice cutscene-like sequence. Once you gain control, beware of the volleys of arrows he fires. Attack him in order to earn hearts, but other enemies take priority. Underworld forces appear part of the way through the chapter, and they're about as formidable as ever. Fortunately, Dark Pit focuses on these enemies instead of you. You'll also receive Recovery Orbs halfway through the section, so don't worry if you lose a bit of health.
Eventually, there will be a sequence where he hijacks an enemy to snipe at you from a distance. Just stay under cover until its safe to come out and attack with charge shots and continuous fire. Worst case scenario, you'll simply take a hit or two without taking it out. Still, this section is fairly uneventful, and you should be able to get through it without too much trouble.
This ground section is unique in that it changes randomly. In short, Dark Pit will be found in two of several locations, and its up to the player to find them. In each location, he will serve as a miniboss, equipped with different weapons. Follow the moving Underworld forces to track him down to one of the following locations.
After the first fight, be sure to grab the food from the center. After all the fights are done, grab the Drink of the Gods before using the Grind Rail to reach the Boss... Dark Pit again.
This battlefield contains a couple of treasure chests, some Underworld forces, Dark Pit, and a Cherubot. Let Dark Pit take out the Underworld forces while you run up to and take command of the Cherubot. From in it, fighting Dark Pit from either a distance or up close should be easy, especially since the Cherubot protects Pit.
While climbing up the mountain, be careful to avoid sniper fire from Dark Pit, who is now equipped with a staff. The treasure chest near the mountain's base contains a Back Shield, so be sure to stop by it. When you get close enough, hit him with a charge shot or two. That will take him out.
Dark Pit will be equipped with a Cannon, so regardless of your weapon, he will be vulnerable to ranged fire. Watch out for his very powerful projectile attacks.
Despite being equipped with a club, it might be a good idea to fight Dark Pit up close, since his charge attacks go through the room's pillars. Alternatively, slowly snipe him from a distance and dodge his large but slow projectiles. When done with him, be sure to investigate the back left corner of the room for a Zodiac Chamber.
Boss: Dark Pit
For his final fight, Dark Pit will use a bow. He will also be capable of flight, so he will be both a maneuverable and powerful fighter. However, he will primarily use ranged attacks, making him a bit predictable. For one thing, he will be vulnerable to melee attacks.
That said, Dark Pit has a small arsenal based off of his arrows. He can shoot several at once, charge at Pit, and use his own melee. If anything, he is extremely similar to Pit using a bow, so it might be worth beating a chapter with a bow simply to get in the mindset.
Dark Pit's most powerful weapon is his rain of arrows. He points his bow to the sky, lights up most of the platform you're on with a red glow, and rains thousands of arrows upon that highlighted area. Naturally, you want to get past the edge of that area before the attack lands. Fortunately, Dark Pit is quite vulnerable during this attack, meaning you can get free shots at him if you dodge in time.