How much information can you obtain from a pulsar-black hole system? Imagine that we have detected an interesting source in the sky that we believe is generated by a pulsar orbiting a black hole.
The challenge here is the following:

What physically relevant information could you extract from the observation of this system?

Note: I am posting an answer with some possible information that we could obtain, but I will NOT mark my answer as the correct one.
 A: The system you describe could be considered analagous to the famous Hulse-Taylor binary system, which consists of a pulsar and a "quiet" neutron star companion in a close orbit.
A paper by Weisberg et al. (2010) describes how a detailed timing analysis of the pulsar alone will yield the masses of both components (to 5 decimal places in the case of the Hulse-Taylor system).
In addition you can of course measure the spin properties of the pulsar - period, spin-down rate, braking index, glitches etc.
You also get the basic orbital parameters - the period, period derivative(s), precession of periastron etc. that allow you to test whether the orbital decay predicted by GR through the emission of gravitational waves is accurate.
Additional tests of GR might come through measuring the Shapiro delay if the system inclination was reasonably high. There are also predicted changes in the pulse shape of the pulsar expected to be caused by coupling between the spin and orbital angular momentum a.k.a the Geodetic effect. There may also be effects caused by the spin of the black hole if the components were close enough together - a.k.a. the Lense-Thirring effect - which might tell you about the spin of the black hole.
A: Well, you could try measuring the gravitational waves emitted from the system. If we assume an order-of-magnitude approximation for the strain $h$ as
$$h\sim\frac{1}{R}\frac{GM}{c^2}\left(\frac{v}{c}\right)^2$$
where $R$ is the distance to the source, $M$ is the sum of their masses, and $v$ is the orbital speed. For a low-mass black hole of about $5M_{\odot}$ and a neutron star of about $1.5M_{\odot}$ with $v\approx0.01c$, then at a distance of about 100,000 light-years, we should see $h\sim10^{-21}$. Obviously, a more massive black hole could increase $M$ by an order of magnitude, but we still would have difficult observing the waves from outside the Milky Way. Inside the galaxy, though, detection would be possible by LIGO. Using numerical models, we could attempt to determine some of the above characteristics using strain measurements. See this LIGO paper for information on numerical fitting.
You could attempt to derive an estimate of the age of the system using the relation
$$T\simeq\frac{P}{2\dot{P}}$$
with $P$ being the period of rotation. This is simply an approximation that holds when the period is much greater than the initial period, $P\gg P_0$.
A: First of all, if the pulsar is not extremely close to the black hole, we should observe almost the usual pattern of a pulsar. Let's start obtaining the distance.
The light that comes from the pulsar may encounter regions with free electrons in the interstellar medium. Those regions introduce a dispersion relation that makes lower frequency waves travel slower than higher frequency ones. If we observe the pulsar in two different frequencies $\nu_1$ and $\nu_2$, we will observe that there are a difference in the observation time $\Delta t =t_2 - t_1$. 
Now, let $DM$ be the dispersion measure, i.e. the free electron density integrated to the line of sight: $$DM=\int_0^d n_e(l) ~dl$$
Light travels a time $t\propto \left( DM\over\nu^2 \right)$, so then we got $$\Delta t= k \; DM \left( \frac{1}{\nu_2^2} - \frac{1}{\nu_1^2} \right) $$ where $k$ is a known constant.
That means that if we know the electron density along the line of sight (for example, if we have other pulsars in the same region of the sky at known distance), we can obtain the distance to the object. Similarly, if we know the distance to our system by other means, we can get the electron density integrated along the sight of view of the interstellar medium.
