Optical chaos based on semiconductor laser (SL) has some vital applications such as optical chaos secure communication, high-speed physical random number generation, chaos lidar, etc. Among various schemes to drive an SL into chaos, the introduction of external cavity feedback is one of the most popular techniques, which can generate chaos signals with high dimension and complexity. For the chaos output from an external cavity feedback SL, a time-delay signature (TDS) and bandwidth are two key indexes to assess the chaos signal quality. In this work, according to the rate-equation model of an optical feedback SL, we theoretically investigate the characteristics of TDS and effective bandwidth (EWB) of chaotic output from a Gaussian apodized fiber Bragg grating (GAFBG) feedback SL (GAFBGF-SL). The results show that with the increase of feedback strength, the GAFBGF-SL experiences a quasi-periodic route to chaos. Through selecting the suitable feedback strength and the frequency detuning between the Bragg frequency of the GAFBG and the peak frequency of the free-running SL, the TDS of chaotic output from the GAFBGF-SL can be efficiently suppressed to a level below 0.02. Furthermore, by mapping the TDS and EWB in the parameter space of the feedback strength and the frequency detuning between the Bragg frequency of the GAFBG and the peak frequency of the free-running SL, the optimized parameter region, which is suitable for achieving chaotic signal with weak TDS and wide bandwidth, can be determined. We believe that this work will be helpful in acquiring the high quality chaotic signals and relevant applications.