Kali Linux can work only with a few wireless cards. Don't try to use a bridged connection or something like that, even if you found a "solution" on the internet, it won't work.
If you are planning only to connect to WiFi connection and don't want to use any WiFi tools from Kali and enable the "monitor mode", then you can dual boot your computer and use the laptop's built-in wireless card. I'm not sure, but if you try to create a bootable Kali USB flash drive or DVD, maybe the built-in driver work again.
If you want to enable the "monitor mode" then you really need an external USB wireless adapters.
And I will show the list of USB Wireless Adapters/Cards compatible with Kali Linux old versions and with Kali Sana too.
If you are still using Backtrack(But really why you need to use Backtrack anymore) then this adapters too will work for you.
Click on the header of Adapter, the name to open Adapter's Amazon page.
Also Read: How to Check the Kali Linux version using Terminal
This is the best wireless B/G/N USB adapter 5Ghz working with Kali Linux. You can see the full specs at it's Amazon page. You don't need to care about the installation of the driver, Kali will automatically detect it. No problems with "monitor mode". This adapter will capture the handshakes even in low signals. It comes with 5dbi antenna, but you can upgrade it.
Also, watch my video on unboxing Alfa AWUS036NHA adapter.
Chipset: AtherosAR9271
More powerful wireless A/B/G/N adapter from Alfa. With more specs.
But why is it in the second place? Because it is not good at capturing handshakes as AWUS036NHA model. Working in "monitor mode". You should not get any problems with the driver.
Chipset: RT3572
If you want a TP-LINK product then this is for you. This wireless adapter has the same chipset which AWUS036NHA have. Working in "monitor mode" and compatible with Kali Linux. You get a good wireless adapter for a low price.
Chipset: AtherosAR9271
In the past, this was the best wireless driver in the list of Kali Linux compatibles. Why? Because it only support 802.11 b/g and have low specs for nowadays. But it is working very fast, very good at capturing handshakes and, of course, working in "monitor mode".
There are so many fake adapters on sale, watch my video.
Chipset: Realtek 8187
Good 802.11 b/g/n wireless adapter, compatible with Kali Linux. Working very fast but again have a problem with capturing handshakes. No problem with drivers and working in "monitor mode".
Chipset: RT3070
Need a very small wireless adapter which will work will work with Kali Linux? Then this is for you. No problem with drivers working in "monitor mode". But if you need a powerful adapter then better to look at Alfa product. This adapter doesn't have an antenna which means you can't get a good quality in use of the long range.
Chipset: RT3070
Good wireless adapter 802.11 b/g/n. Long range and working fast. But there are some problems with the driver, sometimes automatically disconnect and crash the computer. Working in "monitor mode", but again with problems.
Chipset: Realtek RTL8188RU
Need a small size wireless adapter from Alfa? Then this one is for you. Again compatible with Kali Linux, working in "monitor mode". With the antenna and you can even change the size of the antenna. Chipset: Realtek RT3070
Maybe there is some other product you can find on Amazon or on eBay which may work with Kali Linux too, but in the above list are the best adapters.
So all you need to look at when ordering a wireless adapter for Kali is a chipset.
Kali Linux recognizes next chipsets:
Atheros AR9271
Realtek 8187L
Ralink RT3070
Ralink RT3572
Realtek RTL8188RU (But with some problems)