in your particular circumstances would be to buy a wireless router with built-in modem and obtain a couple of USB wireless adapters - always assuming that you have a spare USB port on each machine. These adapters are inexpensive, very easy to install, and work well. You simply install the software, plug them in, and away you go. In fact you don't even need to buy a new wireless adapter for PC 'A' if you don't want to. It already has an ethernet network card, and you could use that as a wired connection from the router (which will have ethernet ports as standard)instead of forking out for a USB adapter.
The important thing to remember with both the router and the USB wireless adapter is that you MUST install the software first - before you plug anything in. When setting up the router you'll need to temporarily connect it to the ethernet port on Machine 'A', but you can disconnect it once the set up is complete - unless you're opting for a wired connection between that machine and the router. In that cased you leave the cable connected. The cable is supplied with the router by the way.