Yes, you may be eligible for gun permit in the District of Columbia even with a prior criminal record.  The Prior Criminal Record can not be felony or any offense (in another State) that carries more than a year in jail.  Furthermore, you can not be under indictment for a crime of violence or a weapon offense.    Furthermore: you can not been convicted within 5 years prior to the application of any:

• Violation in any jurisdiction of any law restricting the use, possession, or sale of any narcotic or dangerous drug;
• convicted of assaults and threats, or threats to do bodily harm, or a violation of any similar provision of the law of another jurisdiction;
• Two or more violations of fleeing law enforcement officer ), or, in this or any other jurisdiction, or Driving Under the Influence;
• Intrafamily offense punishable as a misdemeanor, including any similar provision in the law of another jurisdiction;
• Violation of any firearm offenses;
• Stalking
• Violation of extreme risk protection order
• Not have been acquitted of any criminal charge by reason of insanity or adjudicated a chronic alcoholic by any court.
• Not have been voluntarily or involuntarily committed to any mental hospital or institution.
• Not have a history of violent behavior.
• Not have been the respondent in an intrafamily proceeding in which a civil protection order or a foreign protection order was issued against the applicant.