I earned this on Legendary with a score of 46600, finishing in 21:36.
Given this, I feel this would either best be done on Legendary with a lot of leeway.
If you don't really feel confident about Legendary, I would recommend doing it on Heroic with a few skulls on. I would recommend using Fog and Catch if you do this.
When the level starts, turn around and grab a BR and Shotgun. When you first encounter enemies, the first one has a Gravity Hammer. Personally I switched the Shotgun for the Gravity Hammer, but if you're on Heroic you could probably keep the Shotgun.
Mainly you just want to take the tower as slow as possible, and let Arbiter tank if you're hurt. I only died once on the way up the tower with Arbiter tanking everything, and I think twice at the top.
Focus on using your BR for non-Elite flood, and the Gravity Hammer (or Shotgun + Melee) for the Elites. You can do a test burst on the Elites though, as sometimes they don't have shields.
When you get to the first 'arch' in the path up the tower (where the spider flood thing runs out of) enemies will drop down behind you. I'd recommend turning around and running back, as they all land in the same spot. I was able to kill all of them with a frag grenade.
If you run out of either of your weapons, I'd recommend switching to dual plasma rifles. They are pretty amazing against the flood.
When you go up the first ramp (on Johnson's side), enemies will jump over your ledge to the ledge below you. I'd recommend stopping to take them out.
The top isn't too difficult apart from making sure you don't get overwhelmed. The enemies will usually only come from one side at a time, so you can camp the ledges for them.
They start on the right (Johnson's side, looking away from door), so I would stand on the ledge and kill them either before they jump (if you have a long range weapon) or as they land.
After a while they switch to the other side, but I found them to be a bit more numerous and with more close range weapons, so I stayed back and let Arbiter handle most of these while I hid behind him.
It will finish up with a few coming on the right, one having a fuel rod.
After the Spark fight, let Arbiter take on the few flood ahead, as your Assault Rifle sucks against shielded flood Elites. Grab dual Plasma Rifles off their bodies.
For the Warthog part, if you're playing on Heroic, don't fret if you don't have a high enough score. On Legendary, I earned over 10000 points during the Warthog part.
Don't go out of your way to splatter guys, but feel free to take them out as you go. Most of my kills were from Arbiter shooting down the Sentinels.
When I did the level on Heroic, I had 18000 points when I reached the Warthog, so with the 2 skulls on you should have at least 20k, so the Warthog part should give you enough points for the par score (24k).