The performance of a dual-hop amplify-and-forward (AF) relay system is studied with the kth worst partial relay selection (PRS) protocol under the joint impact of feedback delay and multiple cochannel interference (CCI) at the relay nodes. In our analysis, the closed-form and asymptotic expressions of the outage probability are derived and the achievable coding gain and diversity gain of the system are accurately revealed. Further, based on the asymptotic analysis we propose a power allocation scheme, which can lead to SNR performance gains of more than 1 dB compared to the equal power allocation scheme. Finally, Monte-Carlo simulation results are presented to verify the validity of our theoretical analysis.