Most credit default swaps are quoted as CDS spread (the fraction of the notional that the protection buyer would pay every year for a given CDS maturity). However the contract that's actually traded is more likely to have standardized running spread and an upfront fee. Moreover, some names on the verge of default are quoted as upfront.
To calculate the sensitivity of a CDS's mtm to a change in interest rates, you perturb the interest rates per your risk scenario, recalculate the survival probabilities from the quoted spread, and reprice the swap. Note that if you perturb the IR and keep the survival probabilities constant, then you'll get a number that won't be as good at predicting the PL from IR change. The IR sensitivity is relatively small. Depending on your environment, you may need to calculate it and hedge it (with ED futures or IR swaps), or may be allowed to keep it. The IR gamma of a CDS is so small that it can usually be ignored.
To calculate the sensitivity of a CDS's mtm to a change in the quoted CDS spread, you perturb the quoted CDS spread, recalculate the survival probabilities, and reprice the swap. You should get a number that is roughly a couple of orders of magnitude larger than the IR sensitivity. The gamma is large enough that you should calculate it under most circumstances.
In addition to the IR and CDS spread sensitivities, you should calculate the P&L from jump to default.
You should look at your risk by tenor bucket, i.e. not assume that CDS quotes for different maturities all move in parallel.