Blimey a crossover cable that is old school. Since Twisted Pair cables became the standard you don't really need them anymore so a regular Cat3 patch cable or greater that has come with every router, hub or switch for donekys years would be fine. I haven't tried plugging in one PC directly to another but Steam does need one PC to be connected to the internet for In Home Streaming to work.
Here's a recommendation though. Check if your router is 1000 BASE-T or Gigabit capable, because if it's an older 100 BASE-T or Fast Ethernet router that will be a big issue in terms of latency regardless what cables you use. Outside of fiber routers a Gigabit router is quite rare directly from and ISP even today. A cheap (and in my experience better) solution is to go invest in a Gigabit capable switch and some Cat5e or Cat6 cables (depending on length you need to run them and if there's anything electrical in the way might want to get shielded cables too). They're quite cheap and something like
this or
this will do the trick nicely as you plug your Ethernet into one port and the other 4 distribute it to the connected devices.
Bear in mind your internet connection quality has no bearing on your stream quality but your internal network connections needs to be up to snuff.