To continue from uness' answer (edit: just seen the OP was very old, but will leave here anyway!) . The greeks will be every element of market risk to which the the CVA is sensitive. Writing in words for celerity:
A CVA is a credit linked option on the underlying instrument. You are sensitive to the credit default- (specifically the swap obligation payment failure)- contingent +ve mark to market of the instrument, hence it is a credit linked option.
Consider the CVA on a receiver swap (receive fixed versus float). you are sensitive to:
- interest rates (rates delta)
- credit default intensity (credit delta)
- interest rate vol (due to optional asymmetric exposure profile)
- credit vol (modelled using stochastic intensity)
- credit - interest rate correlation (if rates are low with credit intensity high your CVA increases naturally, so you are short this correlation on this position, and in this case short credit and interest rate vol).
- there is also potentially material credit / rates gamma. You would probably be short credit gamma (the wider the intensity, the more the risk is priced in), much as you would in selling a CDS, depending on potency of correlation.
So - the deltas can be hedged with linear instruments, credit vegas will also be partially hedged with CDS (buying a CDS is short credit gamma), rates vegas are hedgeable. Credit-rates correlation, well, you probably just wear that.
Just a quick verbal overview, but hope that helps.