To distil, and create rum, you must head for Liberty Bay. Make sure you bring with you a fire bug, a sickle and a lot of vials, or cups. Not buckets, as they will not be able to be used together with the Distilling Machine.
Once on Liberty Bay, you will most likely find plantations of sugar cane.

There, you will have to use your fire bug on the useless leaves of the plants. Always remember though, that fire bugs may explode, leaving you without the most important tool of the progress. It also inflicts a harming attack, causing five points damage on you.

Once as the leaves are burnt down, you can slice it down with the help of a sickle. Sickles can be hard to find, and can currently only be looted from witches. When you have made your harvest, you must find a distilling machine.

These can be found on Liberty Bay, most likely near the fields, at Evan’s shop, where you also can buy fire bugs. Using the distilling machine is quite simple; you only have to ‘use’ the bunch of sugar cane on the machine, and then you must be quick when filling your vials or cups, as you will only have ten seconds on you.
Then, it will be empty, and requiring a new bunch.

And there you have it, folks. The Distillation of Rum, the holy drink of the Pirates. Yo-ho-ho and a bottle of rum! Harr!