Purify mode is playable via the "Effect Modes" main menu option and is a 3 minute timed mode.
To clear infected blocks (which are coloured purple in-game), you simply clear lines (it doesn't matter if said lines contain infected blocks or not - though if a line clear contains infected blocks, it helps advance the total faster). Combos (line clears with consecutive tetrimino drops) reward you with extra infected block clears.