Here are my thoughts on the relative difficulty of this game's missions on veteran. I will also include a few tips that helped me.
I will structure my solution like this: NAME - DIFFICULTY - TIPS/INFO
Prologue:
F.N.G. (Training) - Easy - No enemies here, so this is straightforward
Crew Expendable - Easy/Medium - Relatively simple, make sure to make good use of cover and keep pressing forward. The dash out at the end can be tricky, but keep trying and you'll get it.
Act 1
Blackout - Medium - One hard part on this one is going up the hill. There is a shed you can use for cover but remember that unless you push forward enemies will spawn forever, so wait for a good opportunity then move up.
Charlie Don't Surf - Easy - I did think that this one was hard, but there is a nice trick that makes it easier. When you get to the room with all the TVs, head through the door and turn right. Run straight on and then look left. The enemies will be coming out of a side room in the back corner. Throw a flash grenade, and then run straight past the enemies into the room they came from.This will stop them spawning, making things much easier. Once your team clears the room you can pretty much just keep walking to complete the level.
The Bog - Easy/Medium - Do this after a hard level (e.g. war pig) and it will seem very easy, do it on its own and it will seem tricky. As usual just keep pushing forward and taking out the enemies in direct thread to you and this shouldn't be too bad.
Hunted - Medium - this mission has some hard sections too it. It's one of the few I would recommend a video guide for.
Death From Above - Easy/Medium - This isn't too bad. You will be put in charge of an AC 130 in this mission. Just take out the enemies as they come and remember that you can switch gun types with Y and shoot while one of your weapons is reloading.
War Pig - Hard - I was stuck on a single checkpoint on this mission for several hours. It is where the tank is moving up the street and you have enemies to your left in a large building and too your right in what appears to be a shop of some sort. I finally did it by working my way up the left hand side, hiding in the side corners, then when a few enemies were taken out, I used a flash and ran forward into the shop. This caused the tank to move up, I got a checkpoint and the mission advanced. After that the level became easier.
Shock and Awe - Medium - You start this level in a helicopter, and it is pretty much chance whether you die at this part, so keep trying and you'll get it. Then there will be some regular gameplay after you're dropped off, and you will eventually meet up with some allies. The part after this really tested my patience. You start on the first floor and the battle is going on across a small courtyard. The problem here is that enemies love to grenade spam you, making your job a lot harder. You need to get into the house ahead and to your right, once there you should get a checkpoint. The other hard part is carrying you wounded comrade to safety. I found this part too be reasonably straightforward, because when you pick up your friend you gain a lot of health and can survive a lot of hits. Just get to the helicopter and you're done.
Act 2
Safehouse - Hard - I personally found this mission very hard. You need to search 3 houses and then a fourth. The problem is that the enemies are constantly infinitely respawning around you, meaning that if you don't push on you will get nowhere. My tactic was to use flash grenades to get closer to each house, then work your way through the house. One annoyance is that enemies continually spawn both inside and outside of each house while you are inside it, making this one tricky.
All Ghillied Up - Easy - This sniper mission is easier than one shot one kill by a large margin. Use stealth to get past the enemies and you shouldn't find this one to be too bad.
One Shot, One Kill - Hard - Oh boy this mission is hard. Not just in parts but generally from start to finish. The hardest parts are when you are transporting MacMillan, and when you are waiting for a helicopter at the end. Even just moving through the buildings can be very hard on this one. For the end part, put MacMillan down (he is invincible when put down), then run towards a brown shed to the side. You can use the cover in this area of the level to your advantage, as you are pretty much exposed near the Ferris wheel.
Heat - Medium/Hard - this one has you fighting down a hill. Certain parts are very hard, but my tip is to try not to get too preoccupied with the gunfight, and instead take out the enemies in direct threat to you then rush on. Your smoke grenades can help you when you are running forward, and you can also bring in airstrikes to help you. Once the helicopter comes at the end, throw a flash then make a dash down and into it.
The Sins of the Father - Hard - This mission can be extremely hard. The part where you chase the son is tricky, most notably when you get to a main road with a number of enemies. You must keep moving or you will lose the son, so throw a flash and run into an alley at the side. The rest of this mission is also hard, so this may be a level to use a video guide for.
Act 3
Ultimatum - Easy/Medium - This one was straighforward. Certain parts (such as when the trucks turn up) annoyed me. For the trucks, I went upstairs in the building right next to where they come from and then shot out of the window from there.
All In - Medium - This one can be tricky, most notably when you have to take out the armored vehicles. Throw smoke, but remember that the smoke takes some time to actually deploy, and in this time you are vulnerable. Try to rush in and plant your C4 then blow it up.
No Fighting in the War Room - Very Hard - This level is made harder by the fact that you have a time limit against you. Take the path to the right at the start, as I found this easier, and throughout the level you should try to take side routes where possible. A video guide is recommended. Bear in mind you need to be quick or the game refuses to give you checkpoints, making this all the more difficult.
Game Over - Easy - If you've got this far, you're practically done and should be able to do this. The vehicle chase section shouldn't pose too much trouble, remember you can use the items in the back of the truck for cover. Then you need to destroy a helicopter, which can be tricky. Once you're past that though, you're on the home straight. The bridge part is the final bit.
Epilogue (not needed for this particular achievement, but included for completeness)
Mile High Club - Hard - I would say this compares most to the level "no fighting in the war room", purely because you are under a time limit and must keep pushing forward. There are no checkpoints, so you must do this all in one go. Make sure you use flashes to your advantage and get a headshot at the last part.
Thanks for reading my guide, hope I helped someone by making this.
One last thing - don't let the game frustrate you. If you're finding it tricky, take a break and come back later, or even play a different level instead. I would say that this is one of the hardest COD games I have played on veteran (beaten by WAW), so don't expect it to be easy. When I commented to a friend that I was finding a mission tough he said "Well, it is veteran, it's supposed to be hard", and that is a very true comment. But, it is doable, so keep at it and one day you'll beat it.