Sierra 117
The first skull of the game is located here, in this area, near where the first dropship comes to drop off its load. To find it, head to the small bunker buildings along the cliff's edge and start walking towards the water, hugging the cliff. There should be a large rock jutting out over the water; the Blind skull will be on it. Pick it up with the R2 button to unlock it.
Crow's Nest
As soon as this mission begins, you can find the Black Eye skull. Look at the top of the first room here; you'll see a large air conditioning duct above you, with a set of red girders below it. If you walk to where the girders meet the ground, at the top of the stairs, you should be able to jump onto them and jump from there onto the duct. At the far end is the skull.
Head down the steps here and arm yourself at the weapons shelves, then proceed onward until you reach a closed door. You can open these by finding the switch and pressing the RB button.
Tsavo Highway
As soon as you exit the tunnel here, you can pick up the Tough Luck skull. Get out into the sunlight, then exit your vehicle and hop onto the building to the left of the tunnel exit, then hop from there onto the pipe beyond it. Walk down a ways until you reach the yellow gate-like thing that blocks you from walking onward. Jump over it, then look downward; you should see a walkway leading out over the chasm. Drop onto it and walk down; you'll see a platform on your right. You can jump across to find the Tough Luck Skull.
The Storm
The Catch skull is here, but it will be very difficult to obtain. It's atop the small silo on the far side of the docks from where you enter this large area, near where the anti-air Wraith sits and fires. Reaching it is way difficult, however, since both of the two Wraiths here must be alive when you get to it; if either one is dead, then the skull despawns. Lastly, the silo is too high to get to by jumping. When you attempt to reach it, you'll also come under fire from the enemy Ghosts. All of this combines to make the task of reaching the skull almost impossible.
Luckily, there is a way to get to it and pick it up without too many problems, although it will require some luck. You'll have to begin in the corridor where you open the door to lead out to the docks. Hopefully your Warthog is still alive; if it is, then enter the driver's seat when you open the door. If not, grab a Ghost here. If neither vehicle is available, then you're probably screwed.
Anyway, if you get into the Warthog, drive down the ramp near the exit door and head for the silo. You'll come under fire from the enemy Ghosts; ignore them as best you can and drive to the far side of the silo so that the Warthog doesn't kill the anti-air artillery. (If this becomes a problem, try to get to this position in a Ghost instead.)
Hop out of the Warthog here and head into the small passage between the shield on one side of the silo and the silo itself; you'll be in a narrow space where no one can hit you unless they basically come in behind you. Wait for your shields to recharge, then drop a frag grenade at your feet. A second after dropping it, jump; the blast from the grenade will propel you up onto the silo where you can nab the skull.
Everything has to go perfectly in order for this all to work, and you will need some practice to make the grenade jump work possible. Possible it is, though, and you'll need to do something like this to get the skull.
Floodgate
Head down the hill towards the town nearby. You can take the opportunity to obtain the Fog skull. This one's easier to demonstrate than to explain, but as you head into town, a Flood zombie will appear on the rooftops to your right and jump over the path in front of you. He'll be difficult to spot unless you're actually looking upwards, though; try sticking your head up just as you clear the first sandbags, and you should see him.
Anyway, if you can manage to shoot this guy with your battle rifle before he jumps over, he'll drop the Fog skull. Sometimes he'll drop it on a rooftop, and sometimes you'll just miss; if this occurs, drop a frag grenade or two at your feet to kill yourself; you should've just passed a checkpoint and it won't take too long to try again.
The Ark
If you follow Johnson, you'll have to defeat a couple of Ghosts near another section of the bridge that extends over the sand. The bridge here can be reached by climbing up the rocks beneath it and jumping up to the bridge itself. If you reach the bridge, head to the right (towards where you had your last fight against the choppers), and zoom your weapon to look out at the last little spire at the end of the bridge. The Famine Skull is here, on the little ledge at the end of the spire. You can't reach it normally, though; you'll need to grenade jump to it. This is more difficult than the grenade jump on the Storm level, since you have to move forward and hit a pretty narrow target to pull it off successfully. Failure to hit the ledge will cause you to have to jump back up to the bridge, which takes a while. If you can't make it, try it on Normal, where grenades will damage you less.
The covenant
The Thunderstorm Skull can be found here while you're in the Hornet. It's atop a walkway on the second tower that you pass by on your way to the third tower. When you hit the mass of Banshees with the anti-air Wraith below them, turn to your right and look up; you should see a walkway jutting out from the structure there. Get above it, hop out of the Banshee, and walk to the edge of the balcony to find your skull.
Cortana
The Tilt Skull is located in this large room here, where Cortana starts gibbering at you, above the fray. You can get it, but you'll need to wait until the fighting has died down. Look around the room; you should see some mushroom-like platforms scattered around the area. It can be difficult to reach them, but if you find the blue arc-shaped display screen in one corner of the room, you should be able to jump to them, then jump to the upper area of the room where the Skull is located.
Halo
The last gold skull in the game is at the beginning of the level, and is very easy to find. Head up and into the cave near the crashed Pelican. When you do, hug the right wall and follow it until you see it curve into a little passage on the right side. Follow that until you find the skull.