After you finish the main story of the game, the speed run challenge will unlock.
You will need to achieve the RED time of each of the speed run challenges.
The good thing however, is that the timer only ticks when you move. Have patience, plan your moves and take your time. If you don't get it on your first try, try another, once you figure out how they spawn it will get easier.
The bad thing is that you may have to play the level a second time, because you only find out the RED time you need to beat after you've already played the level.
Down below I posted a link from True Achievements where you can already know the times first try!
If you are looking for times for the RED look no further than the walkthrough on Trueachievements : SUPERHOT Walkthrough