I would just like to add to the previous guides, because you can save A LOT of time by cutting out all the unnecessary fluff. I just personally confirmed all of the below in my successful run for the skull.
1. You don't need ANY of the checkpoints, not even the "dead marine" checkpoint, and not a checkpoint anywhere near the skull room, checkpoints are in NO WAY related to the skull.
2. You only need one thing, for the jackal snipers to spawn in "sniper alley".
3. These spawns are what determines if the skull spawns (1 in 7 chance)
4. These spawns happen near to, but are still separate from the "dead marine" checkpoint.
5. I don't know why people suggest crouching on the building before the pelican, completely pointless and unnecessary
6. Picking up the skull is all you need, you do not need to kill any elites after, or even survive
7. No, none of the recent patches have changed this skull (Maka suggested they changed it to a 100% spawn rate, which is false. It is still 1 in 7).
So given the above, you can save A LOT of time by stopping your full-mission restart attempts and following all the silly checkpoints. Instead, although it has nothing to do with the skull, for time's sake I would follow standard procedure until you get a checkpoint after the pelican takes off. This is where you will be restarting from with each attempt, instead of from the beginning of the level. If possible, have this checkpoint happen on the other side of the bridge to save you time making that annoying bridge-to-roof jump with each attempt (checkpoint is finicky though after you leave the other building).
Then, go towards the dead marine checkpoint, but don't look down at it or try to actually get the checkpoint. You want to stand farther to the right, near the edge of the roof, so you can view the jackal spawning behind the large crate on the ground. This is how you know all the jackals spawned (and where your 1 out of 7 chance for the skull happens). As soon as you see that jackal spawn, you are all clear to go as fast as you want toward the skull to check if it spawned. I just go the most direct path I can and go invisible for part of the way to avoid enemies (though you can kill enemies on the way if you want). If it's not there, reload from the pelican take-off checkpoint and try again.
The key is to NOT get any checkpoint between the jackal spawn and the skull. Because the 1 in 7 roll is at the jackal spawn. If you get a checkpoint after that, the 1 in 7 roll is permanent and you will need to restart the mission. Now, how do you avoid getting another checkpoint? Checkpoints do not happen during melee attacks, so melee all the way from the jackal spawn to the skull. If it's not there, reload pelican checkpoint. This will bring your attempts down to more like 30 seconds per run, rather than 5 minutes per run doing the whole mission and following pointless checkpoint routes. So even if it takes you 20 attempts if you get bad RNGs, it will still only take a few minutes, rather than hours doing it using the traditional methods.