Overcoming addiction is difficult and certainly not something you should be ashamed of. But it is also not something you are obligated to share with everyone, especially the stranger who is interviewing you.
If the interviewer asks you about a gap in your resume, you can tell them that you were suffering from an illness but that you are fully recovered now and eager to get to work. You can also say that you were dealing with a family crisis that has luckily been resolved. This allows you the opportunity to be honest, without oversharing information. Most interviewers won’t pry any further, and if they do, respectfully say that you would rather focus on what you can bring to the job. Keep your head up high and stay focused on how you can be asset to the organization!