I use the plastic tray that comes with the ice chest to freeze water - that gives you a block of ice that will usually last me two to three days depending on outside temps, the quality of the cooler, how often you open and close it, etc..
I fillet the fish, then put them into ziploc bags. I then put the block of ice on top of the bags of fish.
I read some advice a few years ago that said you want to be careful doing this because you can actually give the fish freezer burn. I haven't run into that problem but I'd imagine that could be an issue if you use dry ice.