What ever insulation and gaskets first.
Gilles: My suggestion is to get a lid gasket. This material is available from home improvement stores. It is available in a roll. You just peel off the backing and stick it around the lid. The next thing is to insulate any areas that you can get to. You may need to cut an access hole in the bottom of a cabinet or something. This is the BEST thing you can do even if you do NOT convert the box. It will keep you ice solid for a longer period of time. Then you need to decide how you are going to use it. If you are a day sailor and you return to the dock for the evening, you can look at the bar refrigeration components. If you spend days at a time away from your power source then you need a DC powered unit. Once you decide on the unit, you will also want to consider upgrading your battery charger and adding more battery power. Check the archives for some ideas. This project is going to cost you anywhere from $150 - 2500 depending on the route you take.
