Getting a criminal record check by using your name and birth date alone is not possible depending on how you look at it.
What you can sometimes do by name and birth date is verify that no criminal record information is associated with your name. But since it’s possible to have false matches (after all how many John Smith’s, for example, are there in Canada?) you cannot retrieve any criminal record information by submitting on name and birth date information.
If verification of no criminal record is all you need then you can consult a local police agency to get this kind of check done. Many police will not perform this service but many will.
The police will be searching CPIC and so long as they do not find any relevant information associated with your name and birth date you can be issued a police certificate indicating as much.
But what if you need your criminal record? In this case the criminal record check maust be done by fingerprinting because with a fingerprint submission is it impossible to unintentionally have a duplicate match.
This is why a pardon and waiver application always begins with submitting fingerprints to the RCMP for certification. If a criminal record is found it will be sent to you and then you can use it for whatever purpose you have in mind.
TL:DR: If you need your criminal record report, fingerprinting is the way to go.