This thesis presents a search for $CP$ violation in the $D^0\rightarrow K^+K^-\pi^+\pi^-$ Cabibbo-suppressed decay mode using an amplitude analysis. New sources of $CP$ violation have to be discovered in order to explain the matter-antimatter imbalance observed in the universe today. $CP$ violation has not been observed in charm decays up to now, where it is predicted by the Standard Model to be very small. This provides a clean environment to look for physics beyond the Standard Model, which could enhance $CP$ violation in charm decays with, for example, the contribution of new particles entering through loop diagrams. This analysis is performed with a sample of proton-proton collisions recorded by LHCb during 2011 and 2012 at centre-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3.0 fb$^{-1}$. LHCb is one of the four main experiments at CERN's Large Hadron Collider in Geneva in Switzerland. It is specialised in the study of $CP$ violation in $b$- and $c$-hadron decays. The $D^0$ candidates are selected from semileptonic $b$-hadron decays into $D^0\mu^- X$ final states. More than 160000 signal decays are studied, resulting in the most precise amplitude model of this decay to date. This amplitude model, built assuming $CP$ conservation, is used to perform a search for $CP$ violation. The result is compatible with no $CP$ violation, with a sensitivity ranging from 1% to 15% on each amplitude. This result is compatible with the Standard Model predictions and is ruling out any large contribution from New Physics processes in the $D^0\rightarrow K^+K^-\pi^+\pi^-$ decay mode. The $CP$ violation measurements presented here are statistically limited and will benefit from the addition of the Run 2 sample collected between 2015 and 2018 at a centre-of-mass energy of 13 TeV, which is expected to correspond to an integrated luminosity of $\sim6$ fb$^{-1}$. Beyond the additional luminosity, the increase in energy as well as various tracking and trigger improvements make this dataset much more powerful than the Run 1 sample. This thesis also presents one of the improvements made for Run 2, which is a more accurate description of the magnetic field of the dipole magnet through the development of a new field map.