Whenever you die it will appear on the screen your total number of deaths. The game counts a run as a death. So in order to achieve this you must die ten times.
You will earn this achievement naturally and quickly .however there are two other achievements related to the number of times you die which require more patience
If you need to speed up the process you can simply load up a new game and allow yourself to be killed or you the 609mm Hand Cannon to kill yourself