Kinda easy and consistent way to get the King of Jerusalem achievement starting as Cyprus. It will maybe need a couple of initial restarts but it doesn't take too long and you just need to play a month before restarting so its no big deal.
Guide to Getting King of Jerusalem Achievement
This strategy revolves around The Ottomans conquering the Mamluks for you. Unlike some other guides I've seen online this won't take a perfect alignment of the stars and huge amount of luck only some basic initial setup.
We will need to switch our religion to sunni. For this you will have to own a sunni province of course. That a reason why you may have to restart a couple of time in the beginning. You need to conquer Ramazan. This is only possible with them having no allies, hence the restarts.
The actual war is quite simple. You have to keep your fort in the beginning so even if they ferry some troops to your starting island they won't be able to siege it. Also you need to make sure to improve relations with The Ottomans. Then 3 other nations, them being Trebizond, Aq, and dulkadir because you will need military access from the to get to Ramazan without landing directly onto his army.
Don't disinherit your heiress too. You will need positive prestige to later scornfully insult some rival of the Ottomans.
Little side note: this is not required but it helps. You can try to get ottoman ruler to start with Administrator or Diplomat personality. This will make getting the alliance with him a bit easier as you won't get as much of malice to opinion because of him wanting your land.
As outlined earlier you will need to switch to the sunni religion to get the alliance with the Ottomans. You will achieve this by conquering Ramazan. Sending missionary with no maintanace to the province (this should switch the rebel type from separatist to sunni zealots, if it doesn't happen in couple of months you can try to harsh treatment them, that usually works, but I never had any problems with that). Ship your army to somebody else's land so you don't fight the rebels when they eventually teleport to Cyprus. After they siege your capitol accept their demands. Now you have to return the province to ramazan by the "Return province" button so you don't anger the Ottomans who most likely desire that land. At this point if you have less than 100 relation with the Turks you should scornfully insult one of their rivals and that should take you over the 100 and make them not hostile with you and wanting to royal marry you. After that you can get alliance pretty easily.
Fabricate claim on a mamluk province close to the ottomans so you can declare war on them in a bit.
War with the Mamluks
After you have secured the alliance with Turks you will declare war on the Mamluks with promise of land. That war should be pretty straight forward. Just assist your ally whenever possible. You can take the 3 provinces you need to form Jerusalem but don't forget to give enough land to the Ottomans that they don't lose any trust. This is kinda important as you will need them later to declare war on venice, also with promise of territory.
Meanwhile you need to fabricate claim on Crete, as you will need to switch back to the catholic faith to form Jerusalem.
War with Venice
After you have cored your new land and stabilized so you don't have rebels, you declare war on Venice calling in the Ottomans with promise of land. Take Crete in the war. Replicate the same steps you did the first time switching to sunni. After you flip back to catholicism you can happily form the kingdom of Jerusalem and you will get the achivement.