We propose a comprehensive set of indicators (including methods to obtain and analyse them) for computational screening of candidate cathode materials for rechargeable Zn-ion aqueous batteries relying on Zn$^{2+}$ intercalation processes. The indicators capture feasibility of Zn$^{2+}$ intercalation and transport within the material, the thermodynamic stability of charged and discharged material structures, electrochemical stability of the cathode material and electrolyte, volume expansion, and energy storage capacity. The approach was applied to well-known cathode materials ($\alpha$-MnO$_2$ and V$_2$O$_5$) as well as some potential alternatives (MoS$_2$, ZrP$_2$O$_7$, MoO$_3$, and FeO$_2$) to demonstrate the screening workflow and the decision making process. We show that selection of cathode materials for Zn-ion aqueous rechargeable batteries is a multifaceted problem, and first principle calculations can help to narrow down the search. Despite us being unable to identify a particularly successful cathode material, tools and techniques developed in this work can be applied more broadly to screen a wider array of potential material compositions and structures, with the goal of identifying next generation cathode materials for aqueous rechargeable batteries with the intercalation energy storage mechanism not limited to Zn$^{2+}$ ions.