The Old Monk counts like a normal boss and doesn't seem to affect your tendency. I won't see not at all, but very little. From what I've seen doing a pure black character tendency invasion build...beating one of your bosses doesn't shift character tendency to white (verified using Northern Regalia damage as the metric). What DOES move you towards white, is being SUMMONED and helping a host kill a boss. Any boss. Shifts you by just a little. Now, killing an invader for your host (YOU have to get in the killing blow) shifts you quite a bit in character tendency, but the bosses shift you just a bit. Old Monk is not an exception. He shifts you a tiny bit like a normal boss...he doesn't count as a black phantom invader.
Also, "regular" black phantoms like the crossbow chick in 3-1 or the twin wanderers in 2-2, they don't count for your character tendency. Only PvP invaders and "named" black phantoms like Mirenda the executioner in 1-1, Scrivir in 2-2 and so on.
Tip- As Snake mentioned..the EASY way to shift your character tendency to pure white is to find someone and use a password to have them invade you and let you kill them a few times. I believe it's 3-5 kills depending on how black your tendency is. Or...get the spell resurrection, which allows you to 'banish' blue phantoms you have summoned- it sends them back to their worlds and makes them human like a successful boss run. A few resurrections and you'll be in pure white character tendency.