This challenge is simply for having a "Special Finish". The first tier of this challenge is for one special move finish, so it naturally could be the first trophy you obtain. When choosing a character you'll choose between two super moves to execute. They are usually the motion of a super move only slightly altered. For example, Ryu can unleash supers by doing two quarter circles forward and then the punch rather than just one.
You must have one bar of super meter in order to unleash one of these moves, meter is built by attacking enemies or being hit by them. Using a jumping heavy kick (R2) paired with a sweeping heavy kick usually builds the meter fairly quickly. Make sure your opponent is low on health before trying to unleash the super move.
This can be done in any mode, so either do it with a second controller in local matches or in arcade mode. You'll automatically have to earn a few of these in order to face Q, so I would suggest simply going for this in arcade mode while trying to obtain that trophy however the choice is yours.