Not necessarily, but it will be helpful. If the other driver pleads guilty to the crime or receives a conviction from a jury, it is admissible as evidence in your civil case. There is a higher standard of proof in the criminal process than in civil cases, so this will be important evidence in your case. You will still need to go through a separate legal process for a civil case because these are two different parts of the law. Importantly, you can even file a civil lawsuit against the driver if they received an acquittal of drunk driving charges. Again, the burden of proof is different.
In a criminal case, law enforcement will need to prove the charges beyond a reasonable doubt. This is a high standard to meet. In a civil case, your burden of proof is that you must show that the driver was responsible for the crash by a preponderance of the evidence. This translates to the fact that it is more likely than not that your version of events is true, and the driver is legally responsible for your injuries. However, injured victims can use evidence of a criminal conviction to prove negligence.
You will not need to wait for the criminal action to wrap up before you begin the compensation process. However, drunk driving prosecutions generally move quickly, so the criminal process may be over before you wrap up the claims process.