Ace all the Cross-Controller Levels
To ace a level, you need to go through the level without getting hit, and thus have to teleport out of a savepoint. This is probably the hardest trophy to get and may require a few tries on some of the levels. It's a bit uncertain if you need to ace the mini-levels too, but those completes when you lose, and are almost guaranteed an ace (if you just lose with the Vita player in Disco Spinner). I definitely do not recommend doing this with other people.
Escape from San Crispin
There are not that many tricky parts. The most tricky part is where a laser is moving back and forth and you have to use the rear touchpad, and simultaneously run. It's a bit hard to multitask that part, but you can just make it too, if you first use the rear touchpad and then run. Instead of highlighting the invisible red plasma, just use the Vita screen as a reference.
This is probably the easiest level to ace, with very few hazardous parts. This should be no problem.
The Pleasure of Treasure
The hardest level to ace. There a few tricky sections. Mainly the part where you have to jump from one wheel to another at the beginning. If you swing and jump at around 4-5 o'clock at the first wheel you should be good. Another hard part is the circular paths with invisible red plasma. Again don't highlight them, just use the Vita screen as reference.
Jailing Time! A Robot Warden Appears
A boss battle, use the chickens to shoot glowing eggs at the giant death robot, so you could get into the space robot to collect your space pirate treasure (did not expect to write that). The robots missiles should be your main focus, since all of the other attacks are pretty easy to dodge or heavily telegraphed. The platforming sections should prove of no big problems.
Grab all the Prize Bubbles in Story Mode
Prize bubbles are the big bubbles in the story levels that gives you prizes such as objects, stickers and costumes. There are a total of 232 of them in the DLC, but the most of them is out in the open, with only few of them being dastardly hidden. A few requires an extra player, mainly due to wormholes, but there also some 2x sections, specifically made for 2 players. There is even five price bubbles in the third level that requires 3 other people (or controllers (for the lonely people)), so do it with a friend and be happy. Below is a guide of all of the bubbles. Yup, all 232 of them.
Escape from San Crispin
The Pleasure of Treasure
Complete the Cross-Controller Pack story
This just requires you to beat all the story levels. The DLC is not that long and quite fun, and pretty easy, so this shouldn't take you that long.
Complete all the Cross-Controller Pack Tutorials
With this pack, you get another 5 tutorials to introduce you to the wonders of making your own levels. These can be accessed through the "me" button in the main menu, and then under "tutorials". They're at the bottom of the list. Note that you don't actually HAVE to go through them, just select them and then skip it by walking to the end if you want. But where is the fun in that?
Here are the tutorials:
A relatively long video on how to do all the regular things on the Vita. If you've played LBPVita, you might want to skip this video.
Remote Tags Sensor:
A simple tutorial showing you how to place sensors cross-platform.
Shows you how to make a contraption, where you move and object via touch on the Vita, and move the same object on the PS3.
Shows you how to control sackboys and other objects through the controlinator... cross-platform!
Shows you how to create wormholes and how to use them to teleport through the level.
Unidentified Feathered Object
Hatch and catch 10 Golden Chickens in Poultry Panic
I would say this would be the second hardest trophy in the game, and it requires two players each with some practise in the mini-game. The mini-game is pretty simple. The Vita player sees a lot of random chickens fly across the screen over some pipes. The player needs to touch them on the touchscreen when they are over a pipe, so an egg drops down the pipe. The PS3 player needs to catch it, before it touches the ground and throw it in a pipe on the left or rightmost side of the screen. This will cause the egg to follow a conveyer belt where the PS3 player needs to use the bouncepads and turn the lever when the timing fits. If the PS3 player does this to both levers a golden chick will be made. The Vita player has to in the meantime keep red chickens from laying bombs in the pipes, since that will disrupt the PS3 player. Turn 10 eggs into golden chickens via this method (you actually don't have to do anything with the golden chicken afterwards), and this trophy is yours.
Never Tell Me The Odds
Perform a perfect run in Space Spinner
In the minigame Space Spinner you'll have to get the PS3 player to the end, without having him loosing a life. The Vita player doesn't matter.
There is a hard way and a waaaay too easy way to do this.
The hard way would be to grab a friend and actually do it. It's hard since the obstacle course keeps shifting around, and turning as the Vita player controls the spaceship.
The really easy way, it to just let the PS3 player be and just fail with the Vita player. This ends the game and the PS3 is automatically transported to Scoreboard, with no harm done.
Play 5 Community Cross-Controller Levels
When you have cross-controller enabled, all of the levels in the community sections are cross-controller, so go nuts. Completing 5 community levels shouldn't take that long, and you can see all the creative ideas other have. Plus, if you're making levels on you're own, you can also find a lot of tutorials.