Finish all levels with highest possible score.

This is best done on co-op unless you have some crazy left-right brain disconnect and can catch a crystal as it falls.

A few things to note.
1. Crystals fall on the side that the enemy is dangerous, so you have to hit the enemy with one character and collect the crystal with the other. Ideally you catch the crystal before it hits the ground.
2. If you kill and enemy and then die BEFORE you get a chance to collect the crystal, start over. The crystals only drop one time per run, so if you miss a crystal drop you cannot achieve perfection.

Level 1 - 230/230
Level 2 - 230/230
Level 3 - 420/420
Level 4 - 230/230
Level 5 - 230/230
Level 6 - 420/420
Level 7 - 280/280
Level 8 - 250/250
Level 9 - 330/330
Level 10 - 150/150
Level 11 - 250/250
Level 12 - 270/270
Level 13 - 370/370
Level 14 - 380/380
Level 15 - 380/380

Play through all regular levels.

Single Player, Two player, High Scores or No Scores. All you have to do is finish the 15 Regular Levels. Easy? Right?
I added links to various walkthroughs if you are stuck.

Level 1

Level 2

Level 3

Level 4

Level 5

Level 6

Level 7

Level 8

Level 9

Level 10

Level 11

Level 12

Level 13

Level 14
The PC version of ibb&obb has one place, that doesn't appeare in other. Also it takes a lot of time to be solved, so here is a short video how to get through it.

Level 15

Collect all crystals from the secret levels.

There are 8 Secret Crystals Hidden in secret Levels

If you find them Single player and can get them great, if not move along and come back. some of these are near impossible solo.

Crystal #1 (Level 2)

Crystal #2 (Level 3)

Crystal #3 (Level 4)

Crystal #4 (Level 6)

Crystal #5 (Level 7)

Crystal #6 (Level 9)

Crystal #7 (Level 10)

Crystal #8 (Level 12)

Find all levelmen and secretmen.

Once you beat the game normally and find all of the secret levels this is all yours.
See "All Crystals" If you are missing a secret level.

Collect a secret level crystal.

Take your pick of levels from "All Crystals"

Finish a level with the highest possible score.

Take your pick of levels from "Full Zen"

Find ubb and bring him to the next Fin City.

King Ubb is located in level 11.

He is found near the enemy that bounces and chases you.
You HAVE to be playing single-player to control King Ubb using the H-B-M keys.
Also you have to reach the "fin city" with King Ubb, which is located in chapter 12, even the Ubb joins you in chapter 11.
(Thanks Shinnizle,Arktic and Link_Z for information)

Bounce a Fin back and forth between ibb and obb 10 times.

Fins are those little harmless balls you see and have probably had fun running into and making them fly across the screen.

There is no specific level to use but a flat area with maybe a wall behind one of the characters would be nice to prevent losing it. A great spot is Secret Crystal 2, since you have to bounce one back and fourth to get the crystal anyways.

Although the trophy does not say so, it is sort of implied that it cannot touch the ground. So you have to play catch with armless mutants.
Get Ibb and Obb spaced out a small distance and have a Fin between them push into it and use the sticks to simultaneously bounce the fin back and forth. Do not jump into it too hard or you will lose the fin as it has some crazy physics to it sometimes. This will take some doing.

Fall through 6 different warps without touching the ground.

From the level select area (limbo) choose Level 3

This is tricky as both Ibb and Obb have to "chase" each other and the screen movement is pretty large.
It is definately possible for you to do it alone (I did) but I would recommend two players (offline preferably) since controlling Ibb and Obb is a bit counter-intuitive.

Solo just Line them up next to each other and run/jump at the portal approx at the same time. Once you hit it the portal hold both sticks hard right and you will just float through, do not try to aim the guys I got it when I just held the sticks hard right once I started the jump into the portal.

Finish a level without dying.

Fairly Straight-forward as far as trophies go. Just finish a level without dying, I would highly recommend the first level. It is pretty simple and easier yet if you just ignore the crystals that fall from the enemies and just focus on survival.