What a waste of a bounty.
I have installed Microsoft XNA framework on my PC and Risk of Rain works now.
A bit of background, I dug into the annals of my memory and read somewhere that Risk of Rain was developed in GameMaker. The developers jumped through many hoops to get this game cross-platform (so it works on mac and linux too). That said, Game Maker requires the XNA framework and should install it if you install gamemaker.
However, that's jumping through hoops yourself. Microsoft offers the redistributable package from their website, and you should be able to find it Here. File Size - 6.7Mb