This white substance is known as efflorescence. It can be caused by lime or natural salts that exist in the portland cement.
Moisture is activating the salts and evaporation is bringing it to the surface of the brick. It will play out over time but that could take years.
Find the source of the moisture and correct it, the leeching will subside and can then be cleaned away one final time.
If the chimney is not capped, cap it with an oversized cap that will prevent blowing rain from coming down the inside of the chimney. Also look for other sources of moisture. Bad flashing to roof, missing or loose mortar between the bricks, cracks in the mortar, soft brick makeup, anything that would allow moisture in to the chimney materials.
If no obvious source of leaking can be found then seal the bricks and mortar on the outside thoroughly.