For external flows about a body immersed in a homogeneous free-stream, the free-stream boundary condition is fairly simple: almost every hodograph pathline begins from the point corresponding to free-stream velocity. Usually, they will also terminate at the free-stream point, since we are usually dealing with isentropic, inviscid flow. If the flow is viscous or contains shocks, there could be a wake in which case the pathlines could terminate at a different point. There are (usually) two pathlines that begin or end at a stagnation point.
The body boundary conditions are much more difficult. To gain some insight, consider the cylinder example from Section 2.1. From Figure 2, we see that the domain of the hodograph solution is an oval-shaped region. The mathematical boundary of this domain is the pathline of a particle infinitesimally above the stagnation pathline.
The flow tangency condition helps us here; it means that the entire surface of the body has the same value for the stream function. To set the body boundary condition, simply take the velocity distribution on the surface of the body, and make it into a pathline. Assign to this pathline an arbitrary value. This, along with the free-stream condition and the hodograph equation, should determine the solution.
Now, if you are given a specific airfoil shape, and asked to find the velocity distribution, the hodograph method may not be much help. You have to know the velocity distribution a priori to apply boundary conditions; however that is what you are trying to find. About the best you can do here is guess the velocity distribution, apply the hodograph equation, and transform the solution back to the physical plane to see what shape you have. Then iterate this process somehow until you get the desired shape.
However, the hodograph method is much better suited to so-called inverse design. Specifying the velocity distribution, and getting the shape from that, is more or less what inverse design is.
The main problem in using the hodograph method for inverse design is analyzing the airfoil for off-design conditions. Unlike in the physical plane, the body boundary conditions change when the free-stream condition changes. Therefore, you must iterate to a solution for off-design analysis.