Exploring highly efficient, and low-cost oxygen reduction reaction (ORR) and oxygen evolution reaction (OER) catalysts is extremely vital for the commercial application of advanced energy storage and conversion devices. Herein, a series of graphene-like C2N supported TMx@C2N, (TM = Fe, Co, Ni, and Cu, x = 1, 2) single- and dual-atom catalysts are designed. Their catalytic performance is systematically evaluated by means of spin-polarized density functional theory (DFT) computations coupled with hydrogen electrode model. Regulating metal atom and pairs can widely tune the catalytic performance. The most promising ORR/OER bifunctional activity can be realized on Cu2@ C2N with lowest overpotential of 0.46 and 0.38 V for ORR and OER, respectively. Ni2@ C2N and Ni@C2N can also exhibit good bifunctional activity through effectively balancing the adsorption strength of intermediates. The correlation of reaction overpotential with adsorption free energy is well established to track the activity and reveal the activity origin, indicating that catalytic activity is intrinsically governed by the adsorption strength of reaction intermediates. The key to achieve high catalytic activity is to effectively balance the adsorption of multiple reactive intermediates by means of the synergetic effect of suitably screened bimetal atoms. Our results also demonstrate that lattice strain can effectively regulate the adsorption free energies of reaction intermediates, regarding it as an efficient strategy to tune ORR/OER activity. This study could provide a significant guidance for the discovery and design of highly active noble-metal-free carbon-based ORR/OER catalysts.