Complete the game on Easy Difficulty
Complete the entire campaign on Easy Difficulty (as mentioned above, the difficulty trophies do not stack so this MUST be done on Easy Difficulty).

Destroy the gas station
Story related and cannot be missed.

During the second mission, you will have to provide sniper fire for Nantz at the gas station. Once you have killed enough aliens, you will be signaled to shoot the gas tank of a truck and blow up the alien space craft, after which the trophy should pop.

Survive the pool attack
Story related and cannot be missed.

During the first mission, your squad will be traveling through backyards and you will encounter a yard with a swimming pool. As you approach the pool, the game will go into slow motion as aliens pop out from the pool and roof to ambush you. This trophy will pop after you kill these aliens.

Call in the airstrike to defeat the Command Center
Story related and cannot be missed.

During the final mission, you will exit the sewers and have to provide Nantz with cover fire while he provides a target for the bomb strike. Keep killing aliens until a cut scene occurs, after which the trophy will pop.

To unlock a video, you must complete the campaign (any difficulty will work as there are locked videos associated with all three). Once you've beaten the campaign, simply select unlockables from the main menu and choose any video to view.

Complete the game on Normal Difficulty
Complete the entire campaign on Normal Difficulty (as mentioned above, the difficulty trophies do not stack so this MUST be done on Normal Difficulty).

Kill 20 enemies with a sniper rifle
In Hard Difficulty (if not all the time) the sniper rifle will typically be your weapon of choice. It is almost always a better idea to hang back from your squad and pick away the aliens at a distance with your sniper rifle than to go charging at them with your assault rifle. While in Hard Difficulty, the aliens can survive quite a bit of assault rifle fire but are still vulnerable to a well aimed sniper shot, making this weapon a must use during that playthrough. You'll encounter 20 enemies very quickly in the game and will be able to knock this trophy out fairly early on in the first mission.

Complete a mission on Normal without dying, failing, restarting, or continuing campaign
As the trophy says, you must complete a full mission on Normal difficulty without once failing or jumping into a checkpoint. If you play on Hard difficulty first, this is easy to obtain in a second Normal playthrough using your prior knowledge of the maps and enemy locations. If you need to start a new file to achieve this, the first mission ends once your squad has made it to the police station

Kill 15 enemies on the HMV turret
There are two times in the game where you have access to an HMV turret (one of the turrets mounted on the armor Humvee looking vehicles). In each part there will be more than 15 enemies around and the turret cannot overheat or run out of ammo, so simply hold down the trigger and let loose on some aliens.

Kill 2 aliens with one Rocket
This trophy is likely the hardest to achieve because there are so few instances in the campaign where you actually have the rocket launcher and there are 2 aliens that close to one another.

I found the best opportunity for this trophy is directly after you first shoot down an alien spacecraft using your rocket launcher. Right after this boss fight, you will climb down a ladder and head down an alley to a small parking lot where aliens will be shooting at you from the roof. After killing these aliens, you head down another alley directly to your left, where two aliens pop out to block your path. They will be standing right next to each other and this is the perfect opportunity to blow them both up.

Complete the game on Hard Difficulty
Complete the entire campaign on Hard Difficulty.

Some general tips:

Don't be afraid to hang back further behind your squad. The aliens will typically fire at your squad and you can keep your distance and pick them off with the sniper rifle.
Always stay close to cover. Your health can disappear quickly in the larger firefights and you always want an escape plan to hide and let yourself heal before continuing the fight.
Go for headshots as the aliens at this difficulty are able to take quite a bit of fire to the rest of the body without going down.

Unlock all the unlockables
There are unlockables for beating the game on each difficulty, so this trophy will unlock once you have successfully completed the game on all three difficulty levels.