CPU
A big issue with TR64. Officially, "450 MHz or better Intel Pentium-class compatible processor" and a P3 recommended. I suggest you will need at least a P3 or K7 @ 600Mhz or above for good results in a simple game such as Mario64, higher requirements for some other games. TR64 has a "CF" (speed hack) option menu that will be very usefull for those will slower CPUs. This functions much like (perhaps even is?) a frame skip.

Graphics Hardware
TR64 is compatible with Zilmar's plugin specification, meaning it can be used with any compliant plugin. At the time of writing, it comes with one Direct3D plugin and one OpenGL plugin. This means you can run TR64 on almost any graphics card with hardware acceleration, although a powerful modern card such as a GeForce or Radeon is highly recommended. Download any additional plugins from the plugin page. There is no software plugin currently available.

Sound Hardware
TR64 is compatible with Zilmar's plugin specification, meaning it can be used with any compliant plugin. At the time of writing, it comes with a primitive audio plugin that should work (badly!) on any windows-compatible sound card. Download any additional plugins from the plugin page.

Controllers
TR64 supports any combination of Adaptoids, Keyboards and even mice for input. Up to four players. Compatible with and requires DirectX8. If you don't have/don't want/can't install DX8 for some reason, the basic plugins will let you use a preset keyboard layout with DX7. Download any additional plugins from the plugin page.