In the current iteration of the game, there are 5 achievements to obtain. This may change as the game gets updated, but it is complete as of this writing.

Edit: Added 2 of the update achievements.

Beginning of the End
Self-explanatory. Load up either of the levels, then spawn in to begin the game.

Hear My Roar!
Press R (default key binding) at any point during gameplay to roar.

Down But Not Out
Play a complete game until your health drops to 0. Army must be Enabled for this, and God Mode must be Disabled.

Paper Airplane
This must be done on the Poligon City map. Approach the airplane on the ground, located in next to the coastline of the map, then press F (default key binding) to pick it up. Left-click to throw the airplane.




Redeemed
This must be done on the Rio de Kaijiro map. Due to the statue being out of bounds, you have to hit it with a Fireball at long range. Fireball is the only power capable of clearing the necessary distance. Position yourself at the location shown in the screenshot, just below the statue, then aim upwards. Due to the camera placement, it may be difficult to arc the fireball properly. Continue making adjustments to your aim until the fireball eventually collides with the statue.

You can move the camera (using the mouse and scroll wheel) while the fireball is in flight to see where it's travelling. This will help you adjust your aim. (You can also enable God Mode to make this task easier, as doing so will give you unlimited energy.)

New achievements have been added as of the most recent patch. These will be added as I learn about them.

Don't Bite the Hand That Feeds You
This must be done on the Kaijuronto map. The Toku Pop building is right in front of you as you spawn, marked with the pink text. Funnily enough, it's possible to get this achievement on accident just by collateral damage from enemies.

Rock and Stone
Press the spacebar (default key binding) to open up the Kaiju Spawner. Select a Stone Titan, then confirm to spawn it. Simply defeat it using any of your powers to get the achievement.