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.
Long, long ago, way back when computer parts looked like a mess of resistors etc those wrist straps were pretty much needed. Now hardly anyone uses them and you don't hear of many static problems.
All i tend to do is touch the PC case and a radiator at the same time to earth it. Then it's just a case of being careful about handling.
It's maybe not the best way but i've never had any problems.
Only thing i'd point out is the anti static bags, they not anti static on the outside. Safer to get the items out of the bag and put them on the cardboard box than on the bag.