The time-delay signature (TDS) and the bandwidth (BW) are two important performance indexes to assess the chaos signal from a delayed laser system. Based on the spin flip model of vertical-cavity surface-emitting laser (VCSEL), we numerically investigate the characteristics of chaos dynamics in a master-slave VCSEL system, where a chaotic signal generated by a master VCSEL (M-VCSEL) under external optical feedback is unidirectionally injected into a slave VCSEL (S-VCSEL). The influences of injection strength, frequency detuning between M-VCSEL and S-VCSEL, and feedback strength of M-VCSEL on chaos TDS (including intensity TDS (I-TDS) and phase TDS (P-TDS)) and BW are investigated. The results show that by adjusting the injection strength and the frequency detuning, both I-TDS and P-TDS of two polarization components (referred to as X-PC and Y-PC respectively) of the chaotic output from the system can be suppressed simultaneously. Through further analyzing the influences of the injection strength and frequency detuning on the BW of chaotic signal, we find that the BWs of both X-PC and Y-PC of chaotic outputs can simultaneously exceed 30 GHz within a large negative frequency detuning range. Furthermore, by combining the evolution characteristics of the TDS and BW of chaotic outputs in the parameter space of injection strength and frequency detuning, the parameter region for generating the chaotic signals with wide BW and low TDS can be determined. In addition, by reasonably adjusting feedback strength, the quality of chaotic signal from the system can be further optimized.