You need to play through all 60 levels of Time Attack but they are the same levels you had to play through in Story Mode. You have infinite lives here. Starting on level 1, you are given a time limit to complete the level. Whatever time is left over carries over to the next level. You will also gain time at the start of each new level (ranging from 1:00 to 3:00). Collect as many power-ups as you can, especially the extra time bubbles. Each of these will add 3 seconds to the clock. Completing/Failing bonus levels has no effect on the trophy - these are only to boost your score.
Do not use continues should you fail.
If you happen to run out of time then don't use a continue but exit to the main menu. The game will ask "are you sure? Your current progress will not be saved" but this is only talking about the current level you are on, not the entire Time Attack progress so far. Start up Time Attack again and hit resume - it will automatically start you on the current level (the one you just failed) and will restart your time.
If you use any continues, the trophy will not unlock.
If you finished all 60 levels without using any continues then the trophy should unlock once you beat the Intruder on level 60 but if it doesn't, then replay level 60. Do not replay any other level. It will reset your progress from that point on. Note this may take a few tries before it registers as complete.