You will unlock all time attack levels after completing the main story line. To find time attack go to extras then select time attack. The map will look the same as it did in story mode with the exception of high scores being replaced by top times.
You do NOT need to beat the top times for each level for the achievement to unlock. Simply play through all levels again to unlock all the time attack achievements.
This achievement will unlock after completing the following stages:
5-1 Back on the Shellraiser
5-2 Streets
5-3 Shredder's Lair
5-4 Shredder's Throne
If you are still missing energy shards to upgrade your turtles you will have enough after completing all time attack levels.
Note: I do not believe you can gather collectables in this mode as there is no collectable counter. When I go to story it will show 4/4 collected but in time attack it just shows a dash ( - ) so don't expect to clean up collectables in this mode. If this is incorrect please comment and I will revise this note.