Step One: Find the Montessori Schools through School Finder:
https://find.myschooldc.org/?programs=19
Step two- look at the historic waitlist data to get a sense of which schools are harder or easier to get into:
https://public.tableau.com/app/profile/aaron2446/viz/MSDCSeatsandWaitlistOfferData_draft/MSDCPublicDisplay
The PP saying that it's easy to get into these schools at 1st grade is right about some schools, but dead wrong about others- You have a good chance in 1st grade at Breakthrough but basically no chance at Latin American Montessori Bilingual (LAMB). As PP mentioned, DC Lottery gives sibling preference, so if one of your kids gets in, they will more than likely pull the other kid in, so pay attention to those numbers.
Step Three- figure out which schools you like and work for where you will live. Some of these schools are more or less Montessori- some are AMI and some are AMA. Some are still working on certification. Note that DCUM is pretty harsh on Montessori- there seem to be some folks here who post a lot and don't like it so there's a lot of bad press here on schools that from my research or personal experience are good schools. There are others that I chose against. It will be harder to do real research from a distance because the best research is seeing the schools, talking to teachers and admin, and talking to parents.
Step Four- Make sure you qualify. You only need an address to enter the lottery. You need to verify to enroll. This year that was April 30th. A letter from an employer won't count. A paystub with a DC address would.
Good Luck!