This achievement is for continuing 5 times from the same checkpoint.
When you reach a checkpoint you will see a ship fly down which has Tony inside. When you die it will automatically start you back at your last checkpoint. The easiest way to get this is to just let enemies kill you 5 times without activating a new checkpoint.
I personally did this on Mission 1 The Hoodlum Dolls Stage 1(Sweet Angel)
. The first checkpoint I got , I let the enemies kill me 5 times in a row which unlocked the Achievement.