Missable achievements: A couple, but only the jump and pill ones are really missable
First 6 achievements - can be received in about 10-15 minutes of playing on normal by yourself
Xavier Would Be Proud! - 10 points
You used a mutant power 5 times in one game
Simply use your Y attack 5 times during a playthrough. Not at all difficult, just remember after using it the first time during a life, the other times will take away 3 bars of your health, so to get all 5 it would take two lives.
Turn Pyro to Toast - 10 points
You defeated Pyro without using any mutant powers
The boss of the first level is Pyro. Simply defeat him using only your standard X attacks and don't use your Y attack. You can die as much as you want in this game, too. But, the best strategy for any boss in this game is to do a few melee attacks, and when they're knocked to the ground, just as they are getting back up, come either up the screen or down the screen toward them and you can get a few more hits in. It's weird, but if you stand in front or behind the boss they usually seem to hit you getting back up, but above or below them they cannot.
Kidnapped - 10 points
You learned that Magneto has kidnapped Professor X
Story achievement, not missable, but attained after defeating The Blob at the end of the second level.
Score Inflation - 25 points
You earned at least 250 points in one game
Playing alone on normal, you can get up to 460 kills, so you will get this by the fourth level.
Grounded - 10 points
You completed Level 4 without jumping
The fourth level is in caves, and the level that starts after you defeat Wendigo. Simply do not hit A until you defeat the white robot-esque boss at the end. The X attack where you jump on fallen enemies is allowed, as are mutant powers and as many deaths as you want, ha.
Welcome to Die! - 10 points
You were welcomed to die by the White Queen
The end of the sixth level I believe, you are going down an elevator when the white witch appears. Her opening taunt is "Welcome to die!" and I always let her say it, but I honestly do not know if you still get the achievement if she does not say it and you just kill her. I think you still can, as the achievement does not unlock until you defeat her anyway.
Largest Party - 10 points
You played in a 6-player match with five other people
Just have 5 other human players in a game at once and this unlocks. Join a game or have one set up, people join on their own quite often, which is nice.
You Fell to Our Trap! - 25 points
Mystique tried to fool you and five other players in a 6-player game
You need to be playing online, and there simply needs to be 6 people in the game when you get to the end of the sixth level that ends with a magneto battle that turns out to be mystique.
X-pertise - 25 points
You completed the game on Expert difficulty
No challenge to this really, you have unlimited continues. And as always, made easier and faster playing in coop.
Super Mutant - 10 points
You took an energy pill
This is probably the strangest achievement. When you start a game, you can pick the English or Japanese version of the game, and a 4 or 6 person build. The pills show up only in the Japanese version, and in my experience, only while playing coop in a 4 person build. So, that's what I've found to work best, but it might work for you in other modes, the only definite option is the Japanese build.
Bamf - 25 points
You completed the game in 25 minutes or less
This is most easily done in co-op on normal. I got it playing with just one other person. It's a very short game, and the more people you have, the quicker it will go.
Master of Magnet - 30 points
You defeated Magneto with all 6 X-Men characters!
When you set up a game, you can choose what level to start on, the last level, i believe it is level 8, is only the magneto boss battle. Simply defeat him as each character, any difficulty, any number of lives, and this will be yours. ONE IMPORTANT NOTE - This was kind of glitchy for me, i had to beat it three times with one character. But, i think it is because you have to wait for the game to end, and all the credits and everything to finish for it to count.