Getting the Earth-centered inertial (ECI) xyz coordinates of your non-GEO satellite at a specific point in time is somewhat easy. Two line elements are good enough for pointing a tracking ground antenna, and more than good enough for not needing to point an omnidirectional ground antenna.
To go from ECI to Earth-centered, Earth-fixed (ECEF) requires a model of the Earth's rotation, nutation, and precession. The Standards Of Fundamental Autonomy (SOFA) code provides functions that can provide this transformation. I would not go the extra step of converting to latitude, longitude, and altitude because you will need to convert back to ECEF xyz.
You will need to convert your ground station latitude, longitude, and altitude to ECEF because the difference between the satellite coordinates and the ECEF xyz ground station coordinates will yield the position of the target satellite, but in ECEF coordinates.
To convert those relative ECEF coordinates to aximuth and elevation, it will help to develop the transformation to (or from) ECEF and local East-North-Up. You might or might not want to account for the Earth's non-spherical shape. The largest factor by far in this non-spherical shape is the Earth's equatorial bulge. This is essentially the difference between geocentric and geodetic latitude. Geodetic East-North-Up is a bit more difficult, but it is doable (with the right software library). Once you've transformed the relative position to ground station East-North-Up, computing azimuth and elevation are relatively easy.
This reeks a bit of do my homework for me, for free!, so I'm not going to finish this off with the exact equations. If this isn't homework, let me know in a comment to this answer.