For this achievement, you simply must defeat Shredder as each turtle - this is to say, you can do it in Story Mode (online OR offline), or in Survival mode, if you are so inclined. Quickplay does NOT work, however.
If you're going to play through the whole game as one turtle, rather than switching out near the end, Mike and Raph are easiest to complete in an online game - they take a lot of damage if you're playing alone.
I would recommend playing as Don for Survival mode, as he takes slightly less damage and has great range.
You should get this achievement right as you've beaten Shredder (hah, initially suggested a certain someone else - see comments) at the end of your fourth Turtle's playthrough.