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The critical brain hypothesis receives increasing support from recent experimental results. It postulates that the brain is at a critical point between an ordered and a chaotic regime, sometimes referred to as the "edge of chaos." Another central observation of neuroscience is the principle of excitation-inhibition balance: Certain brain networks exhibit a remarkably constant ratio between excitation and inhibition. When this balance is perturbed, the network shifts away from the critical point, as may for example happen during epileptic seizures. However, it is as of yet unclear what mechanisms balance the neural dynamics towards this excitation-inhibition ratio that ensures critical brain dynamics. Here we introduce a simple yet biologically plausible toy model of a self-organized critical neural network with a self-organizing excitation to inhibition ratio. The model only requires a neuron to have local information of its own recent activity and changes connections between neurons accordingly. We find that the network evolves to a state characterized by avalanche distributions following universal scaling laws typical of criticality, and to a specific excitation to inhibition ratio. The model connects the two questions of brain criticality and of a specific excitation/inhibition balance observed in the brain to a common origin or mechanism. From the perspective of the statistical mechanics of such networks, the model uses the excitation/inhibition ratio as control parameter of a phase transition, which enables criticality at arbitrary high connectivities. We find that network clustering plays a crucial role for this phase transition to occur.