What you essentially need is a USB port on PC-1 that will act as a USB device for PC-2.
That is not possible for the vast majority of PC systems because USB is an asymmetric bus, with a host/device (or master/slave, if you wish) architecture. USB controllers (and their ports) on most PCs can only work in host mode and cannot simulate a device.
That is the reason that you cannot network computers through USB without a special cable with specialised electronics.
The only exception is if you somehow have a PC that supports the USB On-The-Go standard that allows for a USB port to act in both host and device mode. USB-OTG devices do exist, but they are usually embedded devices (smartphones etc). I don't know if there is a way to add a USB-OTG port to a commodity PC.
EDIT:
If you do not need a keyboard before the OS on PC-2 boots, you might be able to use a pair of USB Bluetooth dongles - one on each PC. You'd have to use specialised software on PC-1, but it is definitely possible - I've already seen a possible implementation on Linux, and I am reasonably certain that there must be one for Windows. You will also need Bluetooth HID drivers on PC-2, if they are not already installed.
On a different note, have you considered a purely software/network solution such as TightVNC?