Death from above
Get a kill with the Gunship.
A very easy trophy, in order to get it all you have to do is get a kill with the gunship. To do that you will need to capture the Gunship Conquest point, in a game of conquest (you will see it by the little icon next to the Conquest point icon) and then this will spawn the gunship. You can then die, and quickly jump in, as there are only two slots available. Being in the gunship is easy, just shoot at anything on the ground and hope to kill it, and this trophy should not take long from casual playing. Also the gunship wont spawn until the enemy gunship is defeated, so make sure you destroy that first.
Destroy the Gunship.
This is the easier of the two silvers, and will require you to destroy the AC-130 Spectre Gunship which flies around the map when the Gunship Conquest point is captured. It is different for every Armored Kill map, but the gunship is the same. To get the trophy, you will need to be the one that blows up the Gunship in an online game. It is not too difficult, the gunship can shoot back but in a jet or a helicopter (recommended) you can easily dodge it and kill the gunship. The Problems will be that others will try and kill steal as most people will be attacking the gunship as it can deal a lot of heavy damage to ground forces. In order to get the trophy you will need to be the one destroying the gunship, and disabling it wont count. You can easily get it with a jet and use it's thermal missiles which should easily hit the gunship, beware the gunship does have flares.
For those who don't know what the gunship is, here is an image... hard to miss really
Get a kill with the quad bike.
This trophy is really simple and straight forward but can drive some people nuts, to get the trophy you will have to roadkill an enemy in the Quad Bikes. They're easy to distinguish quad bikes, which are at every command point, and in multiple numbers. They're to date the fastest land vehicle in Battlefield 3 and are very easy to control. The problem is to get this trophy you have to drive over the enemy, which wont be easy. Shooting one from the back wont count.
For those who have no idea how they look they look like this:
Win a round of Tank Superiority.
Probably the easiest trophy of them all, all you have to do is win a round of Tank Superiority, the new multiplayer mode available with the DLC. This should not take long, as all you have to do is be on the winning side of the mode. To win you need to capture the middle point on the map, and defend it from enemies until the tickets run out. If the enemy captures the point, your team will have to recapture it or else the enemy will gain more points. The best strategy is to use the tanks and go in teams, as going on foot or on your own in a tank will result in getting overwhelmed by the enemies. The mode is available to play on all four of the new maps. This is the first trophy you should go for, as it is the easiest, and afterwards you wont need to play this mode again (as I recommend conquest to be the mode you use for the remaining four trophies). You do not have to play the whole match, and if you need to, you can be a coward and switch teams half way through the game, or near the end in order to win, but this may not always work if teams are full.
Get 10 kills each with Tank Destroyer and Mobile Artillery.
For me hands down the most annoying trophy in the DLC set, this trophy requires you to get 10 kills with the Tank Destroyer Vehicle class and 10 kills with the Mobile Artillery class. Both are new to the game with this DLC and the vehicles are all available on the DLC maps. The first lot, getting 10 kills in tank Destroyers should be a piece of cake. It is very easy to achieve, by doing just casual tank killing, and shouldn't take too long. The other one is more annoying. Mobile artillery's work weirdly, they have only one seat for both driver and shooter, and you have to switch between them. You will need to aim from a distance in order to actually hit something, or you can roadkill enemies. The artillery is very slow, and very week, it dies from two tank hits. Also they spawn differently on each map, and are very hard to get. The aiming takes a while to get used to but the easiest is to just keep shooting at conquest flags from a distance. This way you can get the trophy easier, as if the enemy is taking the base he wont see it coming. You have to get 10 kills with the two vehicle classes, and they accumulate from both US and RU versions, so if you use the US Version to get 5 kills in tank destroyer and 5 kills in RU, then 9 kills in US and 1 Kill in RU artillery, you get the trophy as long as they both add up to 10.
BM 23 (RU)
I would like to make a small suggestion for getting kills with the Mobile Artillery. If you are decent with the gunship and can unlock the Anti-Air Missiles for the Mobile Artillery, getting the 10 kills is a lot easier. Also makes getting the service star a lot easier. Just a little tip I ran across last week that got me the trophy in no time.
For those still stuck here is a video: