When building a PC and clipping the anti-static wrist clip to the metal part of the computer chassis, does the computer itself have to be on any particular surface?
No, but preferably not cloth.
Is it ok to just clip the anti-static wrist cord to the computer chassis as it stands?
Technically the computer needs to be grounded, usually by having the PS plugged in with the master switch on the back turned off. Careful with the carpeted floor, that will generate static as you walk. A lot of grounding wrist straps are connected to a plug that allows the strap to be grounded to the third prong of the power outlet.