Therefore, if I read your suggestions correctly, the easiest and most cost effective way of me to do this would be via a standard RJ45 cable then
Not quite, you need a CROSSOVER cable, like in the link I posted earlier, provided the rj45 port on the desktop isn't used (if your "router" is actually an ADSL Modem connected to the desktop via a USB connection) and you wish to connect directly to the desktop.
If your modem is an actual router, and should have spare rj45 ports (it ain't a router without them) then connecting the laptop to the router with a standard rj45 cable will work and allow the laptop to use the internet connection as well.