I've managed to get Cool 3 times so far, though every time I've done it I followed the master's timing exactly with no changes... very weird.
Also, I recommend not bothering to go for Cool until you've unlocked the yellow hat, because unless you beat a stage with that hat on Cool mode the record shop won't unlock that stage's music, which is required if you want the final trophy.
Another trophy hint: you have to beat Vs. Mode on every difficulty if you want the last record, so if you're having trouble with it try following the current beat exactly as you're supposed to rather than attempting to freestyle. You won't lose or gain points if you do it correctly, and the CPU will tend to mess up its improv often enough that you'll win by default.
What do I need to do during cool?
You just have to press the buttons in a rhythmic fashion, and you don't have to worry about timing your presses out to make him say full sentences (although it sounds cooler that way).