Redbeard Frank Needs a Drink!
Speak to Redbeard Frank and tell him you are in search of treasure. He will tell you where you can find the treasure, but there is a price to pay. Redbeard Frank is a pirate, and a pirate needs his rum, right? He wants some rum but not just any, it has to be made on Karamja Island. Accept and tell him you will bring some rum.
Now head south down the dock until you reach the men in the blue suits. Speak to one of them so you can sail to Karamja for 30gp. Once you reach Karamja head west until you reach the Karamja Spirits bar. Purchase a bottle of Karamjan Rum for 30gp.
Sneaking the Rum Past Customs
Head to the hut between the bar and the ship. Speak to Luthas whom runs the banana plantation.
Ask him if he could offer you a job and he says he can work something out. He has a crate just outside his door and it just needs to be filled up with bananas. Head to the plantation which is just to the west and pick 10 bananas. Right click on the crate and choose the option "Fill Crate" then stash the rum in the crate by using it on the crate. Speak to Luthas once again and tell him you have filled the crate. He will then pay you 30gp. Speak to the customs officer and ask to sail on the ship back to Karamja. She will search you then let you get on board.
Collecting the Rum and learning of the treasure.
Head to Wydin's Food Store which is just west of the dock.
Speak to Wydin and ask him for a job. He will tell you that you must have a white apron to work there because of safety regulations.
Head to the fishing store which is the next building north of the food store. Grab the white apron from the northern wall.
Put the white apron on and head back to Wydin. Ask him for a job yet again and tell him you have a white apron. He will give you the job; go in the back and search the crates for the rum. Once you find the rum take it back to Redbeard Frank. He will hand you the Chest Key.
Finding the Treasure
Head to the Blue Moon Inn which is in Varrock.
Climb up the stairs and head to the southwestern room. There is a chest in the room.
Open the chest and you will receive a pirate message.
Head to Falador now by any means you feel like. Collect a spade from a nearby house then head to the x which Saradomin points to.
Dig in the spot and a farmer will attack you. Kill him and then dig again...