Butyl Rubber - effective, low cost, easy to use:
Choose any three :-).
Butyl rubber sheet, as used for roof waterproofing and similar, is cheap compared to "proper" antistatic mats. The rubber contains carbon black which provides the conductivity.
Here they shipped large bales of it with a wrapper sheet on the outside which they sold off very cheaply. I bought several square meters of it and it has served very well.
Any sort of conductivity at all will work. 1 megohm per square is fine.
Care! - Very low resistance material is potentially (pun intended) dangerous as it can short to equipment under test or repair. Steven's metal sheet could be very exciting in some cases :-).
I understand that some linoleum flooring works OK. As acceptable resistance can be so high as to be hard to measure, testing with a very simple electrostatic generator would work - as simple as some materials which allow static charge to be produced when rubbed together. If it will discharge electrostatically charged items almost instantly it should be acceptable.
Actually, connecting to the case is probably better unless you also connect the case to earth ground. The easiest way to ground the case is to plug it in, but you don't want to do that until you're ready to power on.
What you're trying to prevent is Electrostatic Discharge (ESD), also known as a static spark, from you to the part(s) that you're working with. Nothing in that equation involves earth ground except that other things that you're likely to touch might be grounded.
So when you're working on stuff, you don't really need to be earth grounded any more than when you're not working on stuff. You just need to be connected to the stuff that you're working on.
An ESD mat will be conductive enough that you can just set the case on it, put the strap on, and be sufficiently connected to the case. Then handle all the other parts by whatever it is that will eventually touch the case anyway, and you'll be fine. No need for earth ground in most cases, though it usually doesn't hurt either.
As a side note, I've built several PC's on carpet without any grounding at all just by handling parts as described above and by casually touching the case a lot. Never had any problems. But I'm also naturally paranoid enough in general to remember to do that.
Best Answer
Yes, the alligator clip should be earthed. As should the work surface beneath the work piece.
It's meant to protect the electronics, not the worker.
The wrist strap does nothing to protect from line hazard, for the user or the work piece.
There should be a 1 megohm resistor between the wrist strap and the connecting wire, so as not to increase the risk to the user if they contact a powered part of the work piece.