How do I file criminal charges against someone who used my identity over the internet to apply for a credit card account?

You can file a police report with your local police department. You should also complete a FTC Fraud Affidavit. The FTC identity fraud affidavit can be used to document and communicate facts related to identity theft in order to formally dispute identity fraud and release an identity theft victim’s responsibility for the debt occurred as a result of the identity theft. To ensure a complete documentation and communication of identity fraud cases, a group of credit grantors, consumer advocates, and attorneys at the Federal Trade Commission (FTC) created a document called Identity Theft Affidavit. Although, the identity fraud affidavit was mainly developed to document and communicate facts related to unauthorized new accounts opened under consumer names, the document can also be used to refute charges on existing accounts.

Although, a police report may be sufficient to document an identity theft case, I think that an identity fraud affidavit is a good starting point for documenting all the facts, evidence and witnesses before contacting the police or the affected companies.