Packet captures are very useful for troubleshooting purposes. The Cisco ASA supports packet captures even in multiple context mode. I normally configure packet captures on CLI level. This can be done by configuring an access-list to match the specific traffic you would like to capture. Add the access-list and the specific interface in a capture command. Mostly I download the capture in raw format for further analysis with a tool like WireShark. The capture can be downloaded via TFTP or via a secure connection (HTTPS) to the Cisco ASA firewall.
When running a Cisco ASA in multiple context mode, I always disable the ability to connect directly to a context for management purposes. That way you have to access the admin context for management access, but this also denies the option to download the capture via a secure connection directly from the Cisco ASA traffic context.
The easiest way to download the capture in multiple context mode is via a TFTP transfer from the system context. Check the example command below. The capture is made within the context named contextA and the capture has the name captureA. The following command can be used to download the capture in raw (pcap) format.
copy /pcap capture:contextA/captureA tftp://10.10.10.10/captureA.pcap
You can now analyse the capture with WireShark