Best I can tell, it's just a "Critical Hit" as most games would call it, a small random chance of extra damage. Apparently certain moves for each character have a chance of getting a clean hit, not all attacks.
From Penny Arcade's Forum (emphasis mine):
A clean hit is a completely random damage boost that can proc on 3 or 4 different moves for each character. These were said to be "signature" moves, ones that they would focus on and probably use a lot. Siegfried's 3b is a clean hit move, for example. There are also some weird ones like Pyrrha's b throw. It's a 20% damage boost and there's currently no way to predict or control it. As Daishi explained it the purpose was to give new players both a guiding light and a better chance to compete. Generally the amount of clean hits will average out for every player over time, but in a smaller setting it has the potential to be very random.
PROC is short for "Programmed Random OCurance".
Corroborating is this answer on gaming.SE:
Clean hits are apparently a pseudo-random chance of getting improved damage. There are some attacks that can increase this chance, and just inputs are said to slightly increase the chance of a clean hit.