What is the pressure inside of a black hole? Is it infinite or is it finite? Does it vary?
There is no pressure that we know of. In fact we really don't know what is inside a black hole (BH).
The classical solutions for BHs have a horizon (or two for the Kerr rotating BH solution) where the internal region is disconnected causally with the external region. In the internal region the spacetime is empty, nothing there, except at the singularity where the spacetime curvature becomes infinite.
Moreover, a person (or particle) going into the horizon (and in the coordinate frame of one going in, or the the particle's frame of reference, it does this in a finite period of time) does not see anything strange going on at the horizon (possible exception later in this answer), and inevitably falls into the singularity, and does that pretty quickly. The gravitational effect that observer experiences inside the horizon gets stronger till it becomes, in the classical solutions, infinite.
The exceptions, or caveats to this story, is that it does not take into account quantum gravity. We don't have an accepted theory of quantum gravity yet (we have some hypothetical theories, like string theory and loop quantum gravity), so as we get closer to the singularity General Relativity becomes invalid and we don't know yet what to have it take over. In fact, there are propositions that there's something called a firewall at the horizon, and everything is destroyed there. There's issues with the conservation of physical information in BHs, and some hypothesis are that the information gets frozen at the horizon, and kept there. This whole issue is in active current research.