Brain-inspired Hyperdimensional (HD) computing is a new computing paradigm emulating the neuron's activity in high-dimensional space. The first step in HD computing is to map each data point into high-dimensional space (e.g., 10,000), which requires the computation of thousands of operations for each element of data in the original domain. Encoding alone takes about 80% of the execution time of training. In this paper, we propose BRIC, a fully binary Brain-Inspired Classifier based on HD computing for energy-efficient and high-accuracy classification. BRIC introduces a novel encoding module based on random projection with a predictable memory access pattern which can efficiently be implemented in hardware. BRIC is the first HD-based approach which provides data projection with a 1:1 ratio to the original data and enables all training/inference computation to be performed using binary hypervectors. To further improve BRIC efficiency, we develop an online dimension reduction approach which removes insignificant hypervector dimensions during training. Additionally, we designed a fully pipelined FPGA implementation which accelerates BRIC in both training and inference phases. Our evaluation of BRIC a wide range of classification applications show that BRIC can achieve 64.1× and 9.8× (43.8× and 6.1×) energy efficiency and speed up as compared to baseline HD computing during training (inference) while providing the same classification accuracy.