To complete the game without using shortcuts, you must beat the game from start to finish, without dying. If you die just once, you have to start all over.
There are 4 regular worlds, with 4 levels in each, with the Temple having 3 levels and the boss level.
To defeat Olmec (the boss), he needs to fall in the lava below. Use bombs or run under him to destroy to the ground. If you run under him, he freezes in the air for awhile then slams and destroys part of the ground. Sometimes when you run under him, enemies will come out of him in yellow fire-ball-like forms. During the fight, you can find various items above you: boxes and treasure.
If you play Co-op, on the other hand, you have a chance to continue if you die. When you or your friend dies, they will become trapped in a coffin in the next level. All you have to do is whip the coffin and they come back to life. I recommend not using a Shotgun or bomb to free them, because they can die.
And this achievement can unlock in Co-op. When my brother and I unlocked the Good Teamwork achievement, he got this achievement as well.