Our brain consists of biological neurons encoding information through accurate spike timing, yet both the architecture and learning rules of our brain remain largely unknown. Comparing to the recent development of backpropagation-based (BP-based) methods that are able to train spiking neural networks (SNNs) with high accuracy, biologically plausible methods are still in their infancy. In this work, we wish to answer the question of whether it is possible to attain comparable accuracy of SNNs trained by BP-based rules with bio-plausible mechanisms. We propose a new bio-plausible learning framework, consisting of two components: a new architecture, and its supporting learning rules. With two types of cells and four types of synaptic connections, the proposed local microcircuit architecture can compute and propagate error signals through local feedback connections and support training of multi-layers SNNs with a globally defined spiking error function. Under our microcircuit architecture, we employ the Spike-Timing-Dependent-Plasticity (STDP) rule operating in local compartments to update synaptic weights and achieve supervised learning in a biologically plausible manner. Finally, We interpret the proposed framework from an optimization point of view and show the equivalence between it and the BP-based rules under a special circumstance. Our experiments show that the proposed framework demonstrates learning accuracy comparable to BP-based rules and may provide new insights on how learning is orchestrated in biological systems.