What would be the radial velocity of a star if we observe the
hydrogen-alpha line (normal wavelength = 6563 A) at a wavelength of
6566 and is it moving toward or away from the earth?

Shift spectrum is due to velocity towards or away from the observer.
Radial velocity of a star superimposes a smear upon that spectrum, one
edge rotating away from the observer and the other edge receding. The
degree of smear is of course dependent upon the relative plane of
rotation.

The shift on your example is so small that a simple proportional
calculation is almost identical to basing the calculations upon the
slightly more complex relativity formula.

The speed of light is accepted by the SI international body as
299,792.458 km/s, coincidentally so nearly 300,000 km/s that it is
good enough for most calculations. Also commonly used is 186 thousand
miles per second.

Your wavelength is longer, so the star is receding, by wavelength
proportion calculation at 137 km/s (about 493 thousand kilometres per
hour or about 308 thousand miles per hour).

Best

>Radial velocity of a star superimposes a smear upon that spectrum

Should that be "Rotational velocity..."?

Ian G.