Hero in the Making
Complete Chapter 2
This is story-related and can't be missed. It will unlock after the cutscene following level 80.
Complete chapter 4
This is story-related and can't be missed. It will unlock after the cutscene following level 160.
Finish the story
This is story-related and can't be missed. It will unlock after the cutscene and credits following level 240.
Complete chapter 7
This achievement may or may not be obtainable. As of now, nobody has legitimately won this achievement or gotten past level 319 in story. While still theoretically possible since the AI can find a route through the level, it remains to be done by a human.
Cut the Cord
Finish a bungee level with three dead allies
Bungee levels are an option in free play mode. Essentially, it's a bungee cord between all players. Since the achievement says three allies must be dead, you have to have four people (or controllers) in the level.
You can do this with three other people, but you can also do this by yourself if you have four controllers. If you decide to the the former, just take turns killing yourselves and letting one person finish. If you decide on the latter, make sure the controller getting the achievement is either the first or last one, turn up the aggressive settings, and keep making random levels until you get something that will kill you at the beginning. From there, make the other three controllers (starting with the one on the opposite end of the cord from the achievement-winning controller) kill themselves with it. Once your three "allies" are dead, complete the level and the achievement is yours.
Score 500k in an arcade game
Refer to Double-Clutch
Score 1.5 million in an arcade game
This is one of the more difficult achievements in the game. Escalation is the easiest mode to get this in, but it's still no cakewalk. The levels in the Arcade are randomly generated, so there is no definitive method to unlocking this achievement, but here are some tips:
First, unlock whichever hero you're best with or you find the easiest to work with. A lot of people prefer the spaceship hero due to its ability to stall at the beginning of the stage, but whatever works for you works for you.
Second, make sure you get all of the coins and don't die within the first 40 levels to maintain your multiplier. Take your time, progress slowly and carefully. While your your multiplier will increase every 20 levels, the levels get significantly harder as well. Earning as many points as possible in the first levels is crucial to getting to 1,500,000.
Third, if you do die or lose your multiplier in the first 40 levels, retry. It's most likely not going to happen without those points.
Fourth, once you do get past level 40, if you come to a level that looks difficult or you question your ability to beat it with all of the coins and without dying, use the slow motion powerup. It can make a lot of difference in a level, and keeping your multiplier is very important.
Die 1337 times
You should easily get this while going for all of the other achievements. If not, play a level where spikes force you from left to right across the screen and set your controller down. Since you respawn automatically, this will rack up deaths for you very quickly.