The programming would be the hardest part, assembling the hardware is as easy as googling "how to build a computer" and then following instructions

Or better yet, get someone who knows how to build one, to show you. He'll be done in 5 minutes or less (probably less, since we're not even putting it into a case).
Getting software to work on more than one computer through a network is pretty hard though, that's where you'll need to find a program that supports such a feature (that, or learn how to use
OpenMPI in your program code, which IMO is even harder than final-year undergraduate mathematics.)
Of course, you can always start with your own PC first, and then when you find it excruciatingly slow you can start thinking about the cluster.