The physical properties of the Zr$_5$Pt$_3$ compound with interstitial carbon in hexagonal D8$_8$-structure was investigated. A set of macroscopic measurements reveal a bulk superconducting at approximately 7 K for Zr$_5$Pt$_3$C$_0.3$ close to Zr$_5$Pt$_3$, also with a correlate anomalous resistivity behavior. However, both the signatures of strong electron-electron interaction, and the electronic contribution to specific heat, increase dramatically with the C doping. For the first time the x-ray photoelectron spectra compared with DFT/PWLO calculations of electronic structure show a complex Fermi surface with high density of states for Zr$_5$Pt$_3$. Also results show the signature of unconventional superconductivity. Indeed, was observed an unusual behavior for lower and upper critical field diagrams of Zr$_5$Pt$_3$C$_0.3$. The temperature dependence of penetration length and electronic contribution to specific heat suggests that electronic pairing deviates of $s$-wave the BCS scenario.