Accurate radial velocities ($v_{rm rad}$) of Cepheids are mandatory within the context of distance measurements via the Baade-Wesselink technique. The most common $v_{rm rad}$ derivation method consists in cross-correlating the observed spectrum with a binary template and measuring a velocity on the resulting profile. Yet for Cepheids, the spectral lines selected within the template as well as the way of fitting the cross-correlation function (CCF) have a significant impact on the measured $v_{rm rad}$. We detail the steps to compute consistent Cepheid CCFs and $v_{rm rad}$, and we characterise the impact of Cepheid spectral properties and $v_{rm rad}$ computation method on the resulting line profiles. We collected more than 3900 high-resolution spectra from seven different spectrographs of 64 classical Cepheids. These spectra were standardised through a single process on pre-defined wavelength ranges. We built six correlation templates selecting un-blended lines of different depths from a synthetic Cepheid spectrum, on three different wavelength ranges from 390 to 800 nm. Each spectrum was cross-correlated with these templates to build the corresponding CCFs. We derived a set of line profile observables as well as three different $v_{rm rad}$ measurements from each CCF. This study confirms that both the template wavelength range, its mean line depth and width, and the $v_{rm rad}$ computation method significantly impact the $v_{rm rad}$. Deriving more robust Cepheid $v_{rm rad}$ time series require to minimise the asymmetry of the line profile and its impact on the $v_{rm rad}$. Centroid $v_{rm rad}$, that exhibit slightly smaller amplitudes but significantly smaller scatter than Gaussian or biGaussian $v_{rm rad}$, should thus be favoured. Stronger lines are also less asymmetric and lead to more robust $v_{rm rad}$ than weaker lines.
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