First U need internet, Get the internet by lan. Then proceed with the steps.
Though, You are using 2016.1, update the sources.list by the following repositories:
deb-src
http://http.kali.org/kali sana main non-free contrib
deb-src
http://security.kali.org/kali-security sana/updates main contrib non-free
deb
http://http.kali.org/kali sana main non-free contrib
deb
http://security.kali.org/kali-security sana/updates main contrib non-free
deb
http://httpredir.debian.org/debian/ jessie main contrib non-free
deb-src
http://http.kali.org/kali kali-rolling main contrib non-free
After that normally install the following ..
apt-get update
apt-get dist-upgrade
apt-get install -y linux-headers-$(uname -r) (Here U
MAY get error, But proceed with the next command)
3.Then, Install synaptic by executing the following:
apt-get install synaptic -y
-Then just open synaptic and search "bcm43142".
-You will see 3 different drivers.
broadcom-sta-common
broadcom-sta-dkms
broadcom-sta-source
-Mark for installation broadcom-sta-dkms and install(or click reinstall) it.
-Restart.
The wi-fi card will show.
Hope it Helped