UPDATED: This will no longer unlock by just selecting Receive Reward on any floor (checked on 188.8.131.52 on 2/19/16.
The Dark Tower (otherwise known as the Infinity Tower) becomes available when reaching Level 13, and can be found in the Great Desert after completing Magic Shore as a solo map (50 gold to enter) or in Skyrock as a Raid (1000 gold to enter). This will only unlock in the Solo version.
In order to unlock this achievement, you will need to complete the 5th floor of the Dark Tower and select Receive Reward to save your progress. Potions are not allowed in the Dark Tower, but scrolls are still available, so stock up on healing or any other type of scroll that could be used as an advantage. Each wave has a 1:00 time span, and if you run out of time (or die) you can use crystals to revive yourself if necessary.
As usual, I suggest using the Archer (or Mage) for this.
In order to unlock this achievement currently, it seems that you just need to complete a floor and select Receive Reward, and you will receive the achievement for completing 5 and 10 floors on the Infinity Tower. This achievement will only unlock if Receive Reward is selected between floors. If you complete 5+ floors, and die instead of claiming the reward, you will not receive either achievement.
Crystals can be used to revive/add more time for a floor to unlock this achievement.
Thanks Van Uden for the information that it works for lower levels.