You will find the Corpser in 'Angry Titan', the final chapter of Act 3. To defeat it, stand near it. It will raise a leg to attempt to hit you. Once it does that, shoot it in the stomach. It will rear back, exposing its entire body. Hit it in the neck, and it will retreat. Do this three times in a row to make it go onto a platform. Then, pull out your Gnasher (shotgun) and hit the two clasps on either side. If you don't have a shotgun, you will need to use another gun or grenades, but the Gnasher is ideal.
After about 45 seconds, Wretches will start to appear, but you should be done or very close by then. If the Wretches start to give you problems, it'll be quicker to restart the whole thing.
This achievement can be done on Co-Op. If done in Co-op, it will not matter who actually kills it. The achievement will unlock for both players, assuming neither one already has it.
The Corpser must be killed on Hardcore difficulty to count. Not Casual or Insane.