The precise determination of the $B_c to tau u_tau$ branching ratio provides an advantageous opportunity for understanding the electroweak structure of the Standard Model, measuring the CKM matrix element $|V_{cb}|$ and probing new physics models. In this paper, we discuss the potential of measuring the processes of $B_c to tau u_tau$ with $tau$ decaying leptonically at the proposed Circular Electron Positron Collider (CEPC). We conclude that during the $Z$ pole operation, the channel signal can achieve five $sigma$ significance with $sim 10^9$ $Z$ decays, and the signal strength accuracies for $B_c to tau u_tau$ can reach around 1% level at the nominal CEPC $Z$ pole statistics of one trillion $Z$ decays assuming the total $B_c to tau u_tau$ yield is $3.6 times 10^6$. Our theoretical analysis indicates the accuracy could provide a strong constraint on the general effective Hamiltonian for the $b to ctau u$ transition. If the total $B_c$ yield can be determined to $mathcal{O}(1%)$ level of accuracy in the future, these results also imply $|V_{cb}|$ could be measured up to $mathcal{O}(1%)$ level of accuracy.
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