Unless it is a fairly modest sum, agreement of a claim for a Collision Damage Waiver (CDW) or Supplementary Liability Insurance (SLI) policy tends to be handled directly between the insurer and the car hire company. You are not usually expected to put the full value of a car on your credit card!
It is really important, especially with SLI claims, that you follow the insurer's process carefully.
In an SLI claim the first part of any claim is on any locally provided insurance your car rental firm has provided. If you only claim on your Supplementary Liability policy you can end up causing real confusion and, in the worst case scenario, risk having your claim rejected.
If your car hire did not come with any liability insurance and you therefore purchased your own primary liability cover then you would claim through that policy.
If you are in any way unsure, simply contact the car hire insurer you bought from and they will make sure you stay on the straight and narrow.
It is likely that on contacting your insurer, that they will ask for details of the incident and documentation to support your claim. This evidence may include some or all of the following so please do retain any such paperwork:
If you have paid for any costs (such as your credit card being debited for a modest claim) you will, upon successful completion of any claim, be reimbursed directly, whilst, if the incident is handled directly between the insurance company and the hire firm, reimbursement will be direct. Larger claims and particularly those for SLI/CDW policies may well be undertaken directly between the car hire company and the insurer.