David,
As you can see from Lubos's comment there is a serious flaw in the formulation of this question, so I will use this answer simply to explain some aspects of the topic. First the flaw:
the energy density of a Schwarzchild black hole
The Schwarzchild solution is a solution of the Vacuum Einstein equations ie $R^{u,v}=0$ and so $T^{u,v}=0$ ie the Stress Energy Tensor is zero throughout the Schwarzchild solution (removing the origin from this manifold avoids the undefinedness there). With zero stress-energy there are no fluids or local energy densities to measure or examine.
What is meant in the question (as it arises from the earlier Stack questions) has some relationships with the Hoop conjecture as Edward points out, but just for clarification I shall add more.
Let us not consider any more an Einstein vacuum and assume that some form of matter (or radiation - I will just say matter below) must have been present "originally". This matter is part of a non-vacuum solution of the Einstein equations, and so there will be a corresponding non-zero Stress-Energy Tensor whose matter is destined to become the Black Hole. So now the question begins to make sense.
So the question is really about what determines whether a given non-vacuum solution of Einstein's equations forms a Black Hole and whether this fact is measurable locally. The sentence that asks this is:
What would be the "signature" of the Black Hole in the components of $T^{u,v}$?$
I dont believe that this answer is known, partly because the space of all solutions of Einstein's equations is not yet known. If one considers the points made below, one might also conclude that GR alone was inadequate to predict the BH formation - matter properties are central too.
There are the classical Hawking-Penrose theorems which give a topological-geometric answer to this question by positing the existence of "closed trapped surfaces", along with certain properties of $T^{u,v}$. In that sense there is an answer, but it doesnt tell us when the closed trapped surfaces will form (generically).
Black Holes arose as physically plausible solutions to Einstein's equations because of the early work of Oppenheimer et al. Here there are two metrics combined to form the matter leading to a Black Hole:
Friedmann Dust (interior) + Schwarzchild (exterior)
(a clever trick to consider "massless dust" allows one to use only the Friedmann Dust solution). These two solutions need to be "glued together" to form the surface of the star. The $T^{u,v}$ (in the comoving frame and its translation into other frames) for this was given in Edward's earlier answer, and is non-zero in the star interior.
What causes another layer of complication in discussing the formation of Black Holes and Event Horizons is the teleological nature of their formation in General Relativity. This arises because "time" is just a parameter in the theory and the formation of the Black Hole is determined by the overall solution (thus in a time independent way). Now it has been concluded that for stellar objects of mass > TOV limit a Black Hole will form. But as remarked above translating this condition into a condition on the Stress-Energy Tensor alone may not be possible.
Physically one might expect that there is some local condition despite all these issues, such as the Hoop conjecture which includes the object's mass in its formulation. There are several subtleties connected with this unproven conjecture and one problem here is that "mass" is not a local property in GR (because mass = energy and the gravitational field contributes energy too, not just the Stress-Energy Tensor - hence we again may need the full solution of $G^{u,v}=T^{u,v}$.)
I shall add this link from Willie Wong for anyone interested in the latest on the Hoop Conjecture.
Finally my answer to the linked question might be of interest.
