You may also use variety of commercial kits. It depends on your task. If you want to store DNA after extraction for a long time, as part of your collection- phenol method is the best one, while commercial kits give you shorter storage period (half a year or so).
Every cell has identical DNA set and you may use any tissiue you like. I use gill filaments and parts of fish fin because it's easy to get and treat.
Regrading the best "organ". The most common way I know of and also use for my research is a fin clip. Just cut a small piece from one of the fins and preserve in 100% ethanol.