What you're looking for actually isn't delta-V. dV is a measure of changed velocity, so to find the dV for making Earth-based life less alive, all you really need to do is find the dV to put the Moon's periapsis under the Earth's surface (or honestly just kinda close would suffice).
What you're really looking for is how much energy it would take to get the moon to that point of crashing into the Earth. Using some well defined physics and orbital-mechanics equations, you could easily calculate the amount of force exerted on the moon by the black hole you've created, then find out how much energy that yields after sustained exertion for a day and a half. This amount of energy, applied retrograde (or not quite retrograde if you're looking for max realism) may or may not de-orbit the Moon. If not, consider relocating your black hole, changing the mass of it (space stations really don't have all that much mass once we get to BH levels), or changing the duration.
Also worth noting is that once a black hole evaporates away, a tremendous explosion takes place, so this might actually be a better propulsive force for do-orbit of the Moon.
MozerShmozer suggested I actually supply the equations I mentioned, and they're completely right! Here is the first Google result for searching "Orbital Mechanics Equations", and it's actually surprisingly thorough. What you're looking for is the relationship between orbital velocity and altitude.
v = sqrt((GM)/r)
Where G is the gravitational constant (a short Google search away), M is the mass of the Earth, and r is the altitude of your orbiting body.
What you need to do is find the velocity of the Moon at its current altitude, then find the velocity at its much less desirable altitude (note it will not be 0, that makes for super not fun math as it divides by 0. Enter the distance from the center of the Earth to the surface), and find the difference of the two. This is your dV (originally sought answer), and now you need to find the amount of energy needed to accelerate an object the mass of the Moon that much.
What this really means, is how much kinetic energy do we need to exert in the opposite direction of the Moon's current trajectory (also known as retrograde)? Luckily for us, someone figured that equation out a long time ago. It's
And we already know both mass of the Moon and both velocities the Moon will be travelling at. So plug all that in to find your dKe and now we have a (astronomically (pun) huge number) that describes how much energy (joules) we need to exert retrograde on the Moon. All that remains is to calculate how much force your black hole exerts on the Moon, and for how long, and then you'll come up with an amount of work done (in joules) on the Moon by the black hole. This should meet or exceed your dKe requirement (also in joules) for a Moon-Earth-impact, and thus the un-aliving of all things currently alive.
Edit 1: Spelling and some clarification.
Edit 2: Followed MozerShmozer's advice of a fuller explanation.