The most common way to diagnose I think it's ultrasonography, when they discover the mole (snow storm) they check for fetal parts. I'm not sure they routinely do anything else for this matter except histological examination which may also show fetal parts in incomplete moles
Anyway if I should go for a choice I'd go with a-Karyotype because it's the most conclusive.
I must disagree about the hCG thing because they don't usually order a quantative hCG usually and the condition is diagnosed first with Ultrasonogram which in most cases will show if there are fetal parts..
that's just my opinion..