The answer is that you can't. Other information and assumptions are necessary.
The angular velocity (which is what I assume is meant by the "rotation rate" here) is given by $v/r$, where $v$ is the tangential velocity around the centre of rotation and $r$ is a radial distance from the centre of motion.
When observing a distant, rotating object, then assuming that rotation axis is oriented optimally, then a spectral line will be broadened by rotational motion. The side coming towards you emits blue-shifted light and the side moving away emits redshifted light. Providing your spectrograph has the resolving power to discern these shifts then they are manifested either by a broadening of spectral features (if the object is spatially unresolved i.e. is viewed as a point of light) or an obvious relationship between position on the object and doppler shift in the case of an object can be spatially resolved.
In the former case, the width of a spectral line yields $2v$, in the latter then we can directly measure the difference in velocity between opposite sides of the object and this is $2v$.
To turn this into a rotation rate requires an estimate of $r$. In general, this is not known exactly for an astronomical object, because it involves multiplying the angular size of an object by its estimated distance, and is therefore a source of uncertainty in the rotation rate. For spatially unresolved objects you can't even do this, so you need some other means of estimating $r$.
A final complication is the angle of inclination of the rotation axis. In general we observe a "projected rotation velocity" $2v \sin i$, where conventionally $i=90^{\circ}$ means the rotation axis is at right angles to the line of sight. Thus one also needs to know $i$ in order to estimate the rotation rate.
