You need to interact (left-click) with an eastern bank booth in Varrock while possessing exactly 6,667 gold coins in your bank account (which does NOT include your backpack holdings). If you do, interacting with the booth will force a dialogue interaction with the banker behind, instead of taking you directly to the bank account screen. However, if those specific circumstances are met, and you interacted with the booth rather than the banker, there will be a hidden dialogue response option "Ask about the Lumbridge Doomsayer¹". The banker will mention a cult, and say that they are "probably dead by now". Instead of acknowledging it, take the option to ask about how to join the cult. The dialogue will be suddenly interrupted, and the banker will suddenly play an animation of him/her putting a robe above his/her head, while an evil laugh sound effect plays in the background. Then, a magic teleport animation, much akin that of any other magic teleport item, will play with both the banker character and your player character, before the player is transported to the new location of the cult headquarters.
Sadly, not much about the area of said cult headquarters is known. However, it is alleged that a top secret, unlisted music track Ian's Got the Longbow will play in the background. This track does not appear on any official listing of music tracks available in the in-game music player, nor have either Jagex or Ian Taylor himself ever mentioned a music titled as such. Another rumour is that, in one of the rooms, lies a lectern, with a scroll open across its top, simply reading "Sector 667". A few players also reported witnessing clearly non-playable characters repeatedly lighting Oak Log fires in a line, akin to how more seasoned players burn many logs in long rows to grind training the Firemaking skill. However, those latter claims are disputed, on the grounds that it could have been another player who decided for whatever reason that said super-secret headquarters were the best place to train the Firemaking skill. On the other hand, very few players have ever successfully visited the cult headquarters, which would make it difficult for two players to meet each other in the same secret location in the same world, which somewhat supports those rumours.
¹ Not to be confused with Doom Slayer, from the Doom series.