An energy harvesting cognitive radio scenario is considered where a secondary user (SU) with finite battery capacity opportunistically accesses the primary user (PU) channels. The objective is to maximize the throughput of SU under energy neutrality constraint and fading channel conditions in a single-user multi-channel setting. Channel selection criterion based on the probabilistic availability of energy with SU, channel conditions, and primary network's belief state is proposed, which chooses the best subset of channels for sensing, yielding higher throughput. We construct channel-aware optimal and myopic sensing strategies in a Partially Observable Markov Decision Process framework based on the proposed channel selection criterion. The effects of sensing errors and collisions between PU and SU on the throughput of latter are studied. It is shown that there exists a trade-off between the transmission duration and the energy lost in collisions.