General Walkthrough
The recommended active skulls on this mission are Tough Luck and Catch. Tough Luck doesn't have much of an effect since the room for the Flood to evade your grenades is limited. As for Catch, the Flood don't use grenades, so it's a consequence-free multiplier.
Please also note that the Flood-infested High Charity has changed a lot since you were teleported there by the Gravemind in Halo 2. It is disturbingly easy to lose your way, so always pay attention. Beating this mission is a matter of measured progression, and the right weaponry is essential. The Brute Shot, Energy Sword, Shotgun, Plasma Rifle and Gravity Hammer are highly recommended weapons against the Flood here. The confined spaces make for several close-quarters battles, so good Melee weapons can always be useful. There are many irritating Ranged Forms that hang from the ceilings, so a medium- to long-ranged firearm is also essential.
Shotguns are also very useful, but ammo is very limited. Use it until you make it to the crashed pelican, get the ammo, and make sure you have an Assault Rifle, SMG, or Plasma Rifle to take care of swarming Infection Forms. Also pick up the Flamethrower to incinerate anything in your way!
Search for Cortana
Drop down and fire upon the Infection Forms. Unless you're actively seeking a higher score, make sure they do not re-animate any of the nearby corpses. Replace your Assault Rifle with a Plasma Rifle and collect some grenades before moving on. Plasma-based weaponry is more effective against shielded Combat Forms, should you encounter any. Throughout this mission, take care not to shoot the Flood Growth Pods on the walls -- they will release Infection Forms if hit with anything other than a direct Flamethrower burst.
As you round the corner, a few Flood Growth Pods will automatically explode. Eliminate them quickly so that they don't re-animate any bodies. Head up the sloped path to reach the chamber's upper level. Find some grenades and grab a second Plasma Rifle to dual-wield with the first. Also replace your Shotgun with an Energy Sword.
When you reach the Porta in the floor, drop through it and immediately take cover in the corner behind you. After a moment or two, Combat Forms will move into the open. Blast them one-by-one with your Plasma Rifle, ducking back into cover if your shields are down or if your weapons have overheated. After the Combat Forms withdraw, Carrier Forms will start waddling towards you. Destroy them with grenades - Plasma and Incendiary Grenades work best. If you can find a Brute Shot around here, replace your Plasma Rifles with it.
Expect to find many Pure Forms and Combat Forms on top of the fleshy hill. The best tactic here is to dash over to the crashed Pelican straight away. The journey there could be a bit fraught, but you can find useful equipment there -- the Flamethrower in particular. Using the Flamethrower make the battle a lot easier, especially on higher difficulties. If you're lucky, the Flood won't follow you as you flee to the Pelican. Camp there, luring Combat and Pure Forms towards you; the latter are easy prey for the Flamethrower.
Be cautious as you leave the Pelican -- Pure Forms and Combat Forms might be still lurking around. Run to the other side of the hill, and make your way to the chamber exit. Note that a slope to the left of the portal leads to an upper level where you can acquire two Plasma Grenades and an Incendiary Grenade. The next corridor is empty, and easy to find your way through.
The room after that contains a small slope and several Infection Forms. The chamber is filled with dead Elites and Brutes, as well as some grenades; replenish your grenade supply, but first use short bursts of flame from the Flamethrower to stop the Infection Forms from taking advantage of the corpses in the room. You can throw a grenade to set off a chain reaction and weaken or kill any corpses that do get re-animated. In the next tunnel, grab either a Gravity Hammer or Energy Sword (your choice), and then drop through the hole to continue.
After you pass through the entrance, head straight to the left. This area provides cover from a group of Ranged Forms as you dismember the first wave of Combat Forms in close combat. Operate from here, drawing the Flood to your position.
Co-op Method
Another way to beat this mission would be to use a better player to escort you in Co-op. Follow them, and don't stop to kill anything -- just run to each checkpoint. Keep the Energy Sword handy if you run into some trouble but you can take whatever weapons you feel like (though you should probably let your more experienced friend have first pick). This only works on lower difficulty modes, though, and you will still die a lot.