Hardness is one test as diamond is 10 on the moh's scale of hardness (out of 10) but I wouldn't recommend it. I don't think there really is a way of doing this without consulting an expert. I took one to a jewellers once and they had a look and told me it was "YAG" or cubic zirconia. I think the way they find out is by examining it to check its refractive index.
thermal conductivity tester, google diamond testers, I once saw an auctioneer use one that looked like a fountain pen with a small wire protruding from it. any help?
Cz weighs more than diamond, its why you see it sold as eqivalent carat weight, so you could weigh it against an similar sized diamond or see if it weighs heavier than the carat weight its supposed to be