Are black holes very dense matter or empty?
The phrase black hole tends to be used without specifying exactly what it means, and defining exactly what you mean is important to answer your question.
The archetypal black hole is a mathematical object discovered by Karl Schwarzschild in 1915 - the Schwarzschild metric. The curious thing about this object is that it contains no matter. Techically it is a vacuum solution to Einstein's equations. There is a parameter in the Schwarzschild metric that looks like a mass, but this is actually the ADM mass i.e. it is a mass associated with the overall geometry. I suspect this is what you are referring to in your second paragraph.
The other important fact you need to know about the Schwarzschild metric is that it is time independent i.e. it describes an object that doesn't change with time and therefore must have existed for an infinite time in the past and continue to exist for an infinite time into the future. Given all this you would be forgiven for wondering why we bother with such an obviously unrealistic object. The answer is that we expect the Schwarzschild metric to be a good approximation to a real black hole, that is a collapsing star will rapidly form something that is in practice indistinguishable from a Schwarzschild black hole - actually it would form a Kerr black hole since all stars (probably) rotate.
To describe a real star collapsing you need a different metric. This turns out to be fiendishly complicated, though there is a simplified model called the Oppenheimer-Snyder metric. Although the OS metric is unrealistically simplified we expect that it describes the main features of black hole formation, and for our purposes the two key points are:
the singularity takes an infinite coordinate time to form
the OS metric can't describe what happens at the singularity
Regarding point (1): time is a complicated thing in relativity. Someone watching the collapse from a safe distance experiences a different time from someone on the surface of the collapsing star and falling with it. For the outside observer the collapse slows as it approaches the formation of a black hole and the black hole never forms. That is, it takes an infinite time to form the black hole.
This isn't the case for an observer falling in with the star. They see the singularity form in a finite (short!) time, but ... the Oppenheimer-Snyder metric becomes singular at the singularity, and that means it cannot describe what happens there. So we cannot tell what happens to the matter at the centre of the black hole. This isn't just because the OS metric is a simplified model, we expect that even the most sophisticated description of a collapse will have the same problem. The whole point of a singularity is that our equations become singular there and cannot describe what happens.
All this means that there is no answer to your question, but hopefully I've given you a better idea of the physics involved. In particular matter doesn't mysteriously cease to exist in some magical way as a black hole forms.
For non-experts it is worthwhile mentioning the simpler 'Newtonian' interpretation of a black hole. Given the fact that most theorists believe that the General Theory of Relativity (GR) is actually inapplicable in the vicinity of a black hole singularity, that viewpoint is also not necessarily disfavored. (This point seems to meet some criticism; please see also my comments below to understand why I am saying this. There is two related problems there: 1) The BH is, first of all, an astrophysical object which has been conjectured to exist based on GR but is not identical to a particular singular solution of GR 2) One can debate about where a BH starts, e.g. at the event horizon, and thus how big it is. In the vicinity of a BH space-time is considerably curved this affects the meaning of distance.)
Already long before Schwarzschild (John Michell 1783) 'dark stars' were discussed. These are objects whose mass concentration is so large that their escape velocity exceeds that of light (John Michell assumed that light was composed of corpuscles as proposed by Newton - an idea which got later temporarily disfavored).
Imagine throwing such a light particle on earth. If its velocity is small it will fall back to ground. However, if it exceeds the escape velocity it will never return. The necessary velocity depends on the mass of earth, if the attractive force is given by Newtons Universal Law of Gravity, but also on the radius of earth since it defines your distance from earth's center of gravity.
This consideration leads to a 'critical radius', below which light cannot leave a given spherical mass distribution $M$ if it has the finite velocity $c$. This radius is $r = 2(GM/c^2)$ and in agreement with the critical radius of the Schwarzschild metric.
In this picture, your question for the density of a black hole is answered simply by $\rho=M/V=M/(4/3\pi r^3)$ with $r$ the critical radius given above. This is the average density of a spherical mass distribution which light cannot escape.
Besides, the term 'black hole' was introduced by John Archibald Wheeler after David Finkelstein recognized that the Schwarzschild metric had this property of dark stars.
Neither GR nor this Newtonian picture describe what happens inside of a black hole. But, based on such considerations, physicists agree that a (non-rotating) mass-distribution compressed to densities above the critical one form something that, what concerns its exterior, is described by the Schwarzschild solution of GR.
Also recall the equivalence of mass and energy. GR treats both forms in the same way. They both curve space-time and get influenced by this curvature. When matter, e.g. some gas, falls in a black hole it gets disrupted and it is probably better thought of as energy (maybe elementary particles but maybe also something that cannot be called a particle).
One of the strongest principles in physics is energy conservation. The matter that falls into a black hole is presumably not destroyed but converted into some form of energy. You can then ask what the energy density of a black hole is. The answer, as above, depends on where the black hole starts. In the case of a non-rotating black hole, a preferred definition is that it begins at the Schwarzschild or critical radius.
The energy of such theoretical solutions of GR that represents a BH is stored in the field configuration, i.e. in the curvature of space time. There is no reference to the state of the matter that formed it. The situation is however that astrophyiscal BHs gain their mass/energy by feeding on matter, e.g. that of a collapsing star which may have initially formed it and later possibly also matter in its environment. How this matter behaves and what happens to it while it is digested (how it is disrupted and what forms it takes on intermediate stages) is not described by GR which is a theory only for the gravitational force. It does already not account for what happens to matter that approaches the event horizon from outside (how it lights up and radiates). Also GR alone does not answer if and how BHs radiate of and loose energy in the form of Hawking radiation.
The question what happens to the matter that falls into black holes can therefore not be answered by reference to GR alone. They can however be addressed within Quantum Field Theory combined with classical (non-quantum) GR. For its ultimate fate, somewhere inside the event horizon, we would also need something like a quantum version of General Relativity.