Actually, I found this part of my arcade build to be quite easy.
What you need:
A box of Male 0.187 Quick Disconnects:

A box of Female 0.187 Quick Disconnects:

A crimping tool:

A Spool of wire (~ 22 - 24 AWG for your button should be fine)
Then, just have a look at the attached image below, essentially you take your computer (fig. 1), take the front panel off of it which will reveal the power button wires, then snip them (fig. 2), wire male quick disconnects to the ends of the wire and run two lengths of wire that will reach to your button and crimp female ends onto those (fig. 3), then crimp another set of female quick disconnects to your button as shown (fig. 4). The black wire should go to the bent prong if you are wiring to a microswitch, and the other wire (may be yellow or sometimes PCs have them in different colors) goes to the lower of the two prongs coming out of the back.
If you are looking to keep the PC intact where you can restore it to how it once was before you started hacking away at it, simply wire some female QDs to the other two ends of the wire that you snipped from the front panel.