In the Washington, DC area, you should start to look at least 60 days prior to your move date.
Most companies in this area require a 60 day notice to vacate and so know availability at least 60 days in advance (occasionally people will give notice earlier). By doing this, you’ll have the best chance of getting the apartment that best meets your needs.
Once you find it, I would also encourage you to lease it right away as in this area, apartments go quickly. Although we’ve seen a lot of specials and availability over the past couple of years, that is quickly becoming a thing of the past. Specials are going away, rents are going up, and apartments are leasing really quickly.