For accurate calculations, convert (lat, lon, elevation) directly to earth-centered (x,y,z). (If you don't do this, you need to retain additional information about the local normal ["up"] directions in order to compute angles accurately at nonzero elevations.)
Given two points (x,y,z) and (x',y',z') in an earth-centered coordinate system, the vector from the first to the second is (dx,dy,dz) = (x'-x, y'-y, z'-z), whence the cosine of the angle made to the normal at (x,y,z) is the inner product of the unit length versions of those vectors:
Cos(elevation) = (x*dx + y*dy + z*dz) / Sqrt((x^2+y^2+z^2)*(dx^2+dy^2+dz^2))
Obtain its principal inverse cosine. Subtract this from 90 degrees if you want the angle of view relative to a nominal horizon. This is the "elevation."
A similar calculation obtains the local direction of view ("azimuth"). We need a level vector (u,v,w) that points due north. One such vector at the location (x,y,z) is (-zx, -zy, x^2+y^2). (The inner product of these two vectors is zero, proving it is level. Its projection onto the Equatorial plane is proportional to (-x,-y) which points directly inward, making it the projection of a north-pointing vector. These two calculations confirm that this is indeed the desired vector). Therefore
Cos(azimuth) = (-z*x*dx - z*y*dy + (x^2+y^2)*dz) / Sqrt((x^2+y^2)(x^2+y^2+z^2)(dx^2+dy^2+dz^2))
We also need the sine of the azimuth, which is similarly obtained once we know a vector pointing due East (locally). Such a vector is (-y, x, 0), because it clearly is perpendicular to (x,y,z) (the up direction) and the northern direction. Therefore
Sin(azimuth) = (-y*dx + x*dy) / Sqrt((x^2+y^2)(dx^2+dy^2+dz^2))
These values enable us to recover the azimuth as the inverse tangent of the cosine and sine.
A pilot in an airplane flying west at 4000 meters, located at (lat, lon) = (39, -75), sees a jet far ahead located at (39, -76) and flying at 12000 meters. What is are the angles of view (relative to the level direction at the pilot's location)?
The XYZ coordinates of the airplanes are (x,y,z) = (1285410, -4797210, 3994830) and (x',y',z') = (1202990, -4824940, 3999870), respectively (in the ITRF00 datum, which uses the GRS80 ellipsoid). The pilot's view vector therefore is (dx,dy,dz) = (-82404.5, -27735.3, 5034.56). Applying the formula gives the cosine of the view angle as 0.0850706. Its inverse cosine is 85.1199 degrees, whence the elevation is 4.88009 degrees: the pilot is looking up by that much.
A north-pointing level vector is (-5.13499, 19.1641, 24.6655) (times 10^12) and an east-pointing level vector is (4.79721, 1.28541, 0) (times 10^6). Therefore, applying the last two formulas, the cosine of the azimuth equals 0.00575112 and its sine equals -0.996358. The ArcTangent function tells us the angle for the direction (0.00575112, -0.996358) is 270.331 degrees: almost due west. (It's not exactly west because the two planes lie on the same circle of latitude, which is curving toward the North pole: see Why is the 'straight line' path across continent so curved? for an extended explanation.)
By the way, this example confirms we got the orientation correct for the azimuth calculation: although it was clear the east-pointing vector was orthogonal to the other two vectors, until now it was not plain that it truly points east and not west.