Getting a Broadcom chipset to work with linux is a royal pain, especially if you have to go the "bw43-fwcutter" route to get otherwise unobtainable firmware blob.
I was never able to get an older Broadcom PCMCIA card to do injection under Arch linux. The Arch linux wiki has a Broadcom Wireless page that has a section for the
wl driver. That page suggests unloading all other possible Broadcom drivers before loading the
# rmmod b43
# rmmod ssb
# modprobe -v wl
You should have another
xterm open doing a tail on appropriate log. I've not run Kali linux, but something like
journalctl -f works on "systemd" based distros, and
tail -f /var/log/messages might work. If the Broadcom chipset is on a USB or PCMCIA device, start tailing the log before you plug it in, and keep tailing until after you do the
Here's a link to Installing Broadcom BCM43142 Drivers on Kali Linux - it looks a bit sketchy to me, however.
Another possible problem is that you're running
airmon-ng against an interface named "eth0", which is usually an ethernet (with a cable) interface. Verify that you're using the correct interface name: do
ifconfig -a to see if some other interface names present themselves.