How may I go about becoming a Drill Sergeant in the Army Reserves?

Easy: career counselor. Request a drill sergeant unit. Pass a pt test. Message me after,

Talk to an inservice recruiter/ re-enlistment NCO. And this is the number to the 108th Training Command. They own the Army Reserve Drill Sergeants. They should be able to send you in the right direction, there is always a need.
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What SFC Christopher Adame said.

Hello Spc Hard. I praise your decision to become a Drill SGT. My fellow still gets will agree that becoming a Drill SGT is one of the most challenging and wise decision we had made. Is not only rewarding as far as career, but you are part of something big. I served with the 108th, I went to Fort Benning for duty. I served with a unique bunch of professionals and it was a rewarding experience. Research where the 108th has units at and find one close to your home, ask them for a pre-qualification checklist and good luck.

Depends on where you are located but the 98th and 95th are pretty much everywhere. I can give you a contact info if you need, but your career counselor would have everyone's contact and can get you a fast meeting with 1sg and Co of your closest company.