As Venice, capture the Holy City of the religion that the city of Venice follows.
Play as Venice on Large Islands on a huge map with max City-States (41). You don't want a difficulty level too low - Warlord recommended. Avoid all World Wonders that grant bonuses to Faith, and the Piety social policy tree. Pick the civ closest to you to maximize the chance they will send a Missionary or Prophet your way, and then establish trade routes with that civilization and City-States following their religion to maximize religious pressure. If you get any Great Prophets while waiting to adopt the religion of your target civ, place them on sleep for later use or use them to create a Holy Site.
Use one of your Great Merchants to buy a City-State near that civilization, preferably on the same island - this will give you a nice staging ground for conquering them later. Once you've got their religion, conquer that civilization.
Interesting side effect - because you never found a religion, when you get to the Industrial Age, you will not be able to purchase Great Prophets. If you decide to add your own religion - after gaining The Great Betrayal achievement - you will need to either use one of the Great Prophets you placed on sleep earlier, or complete the Piety social policy tree completely so you will gain a Great Prophet (which can then be used to found a religion).
This achievement most likely refers to the Fourth Crusade, where Venice's merchants provided a fleet for the crusaders, and then diverted the Crusade towards the Byzantine Empire instead of the Holy Lands, leading to the Sack of Constantinople in 1204.