You will encounter the Chimera during the "Spider and the Fly" mission in the opening section of the expansion with President Myers. The Chimera is a very large mechanized unit, similar to a four-legged tank. You will get to a section where you are ambushed (the Chimera is under the canvas in the center of the room) -
this is not where you will get the achievement - the objective after the ambush is "Flee the Chimera" when it wakes up, so just follow President Myers' instructions and follow her to get away from the Chimera. For now.
Once you've escaped, continue with the mission until you have to go up some stairs and wait for an elevator to arrive. The Chimera will show up again, and this is also
not where you will get the achievement, so just follow the objectives until the elevator arrives and you can leave. You'll then come to a scripted sequence where you have to shoot a cable, jump, and grab cables - do this. At the end, you'll fall and be knocked unconscious, and the Chimera falls with you.
After the next cutscene when President Myers revives you, the Chimera will wake up -
now is when you get the "Defeat the Chimera" objective for the achievement. There will be three objectives - first to shoot the weak spots of varying sizes, which will show as an orange-yellow color, then second to optionally take out the drones it will launch. The final objective is to jump on it's back when prompted and hit the appropriate button (
with a controller) to finish it off. The achievement should pop here.
There will be plenty of ammo in the room, as well as a couple of (not automated) floor turrets in two of the corners. For the turrets, you have to remove the gun and carry it with you. There are also several security turrets on the ceiling you can hack to your advantage. Just stay on the move as much as possible, and damage it while it's reloading and avoiding it when it's not.
Once the fight is over, you have the option to grab its core, which is lying on the ground between the Chimera and where President Myers is standing to continue the mission. You can use the core for crafting a weapon mod, or as a display in V's apartment.