How does one calculate inclination from a graph of radial velocity? If a binary star system has a non-zero inclination, how would that show up in a graph of radial velocity?

How could I then extract the inclination from the graph?
From its RV graph alone, with no other information, you cannot calculate inclination.  This is why RV velocity measurements are typically reported as "$v \sin{i}$", because what you're actually measuring is the orbital velocity projected along the line of sight.  Without other information, you cannot disentangle the orbital velocity from the viewing angle.

What additional information could be had that would allow you to disentangle?  The following list is probably not exhaustive.


If you can observe the two stars as separate objects, then you can track their relative positions through an orbit and can measure inclination that way.
If you have "light curves" (brightness measurements over time) for the star(s) and you see periodic eclipses corresponding to the period and correct epoch from the radial velocity data, then you know the stars are aligned such that they periodically have their light blocked by one, the other, or both.  In such a case, you know that the inclination must be close to 90 degrees else the eclipses could not occur.
If you have reliable distance measurements to the stars and determine somehow that they are main sequence stars, you can determine the star's masses via theoretical stellar evolution models and calculations given a measurement of absolute luminosity (from a measurement of the apparent luminosity coupled with the distance measure).  Once you know the star's masses, you can calculate the expected orbital velocity, compare with your $v\sin{i}$ measure, and determine the inclination.
If the stars are not main sequence stars in the above scenario, then you'll need some additional information (their ages should be sufficient) to correctly extract a mass from their apparent luminosity and color.