I want to play the original Dead Rising on my PC - How would I go about doing this?

You don't. It was an Xbox 360 exclusive. However Dead Rising 2 and Dead Rising 3 are on PC. If you would like to play those.

I know it doesn't have a port, I'm asking if it's possible for me to play it on my PC somehow. Using some emulator, for example.

Nope. there is no existing Xbox 360 emulator. Nor there is a PS3 either. its going to take a few years after the specific console is dead for it to be emulated. For example the PS2 died around 2006 due to the release of the PS3. It took around 2010 or 2011 for an emulator to finally come through. On TOP or the life-span that the PS2 had lived which had the longest life-span. In console history Reaching about 12 years.

Damn, so I'm gonna have to wait awhile.