Every time I see a cleanly isolated figure in somebody's graphic I ask myself the same thing while sobbing with jealousy. The truth is that the overwhelming majority of all my isolated images either comes pre-cut (such as the Dangan Ronpa sprites) or has very convenient backgrounds that are removed easily.

Take Crowley, for example:

For the most part he stands out in the shot, especially once you add some brightness and contrast. Then it's easy to use a layer mask and fill that background away and touch the rest up with a soft brush (I'll get back to that in a bit).

I'm usually hugely lazy about it though and depend on a corresponding background and distracting textures to hide my sloppy work. Case in point:

SO. While I'm far from having the secrets to flawlessly clean cuts I can recommend a few things:

Use a layer mask to cut your image. You work on it in black and white (black being everything you want deleted, white being the rest) and you can easily remedy borders if you've gone to far. You can also use the fill tool on it to get bigger areas out of the way and cut along the border easily if it's one of those convenient screencaps I mentioned.

[Click for bigger version]

If you want to be thorough I recommend using a background colour that clashes with your image, so that you can see any spots you missed immediately. I also recommend changing the background colour (mostly making it lighter and/or darker) before finalising your cut, so that you can spot any mistakes you might have missed.

For details I zoom in as needed and use one of the default Hard Round brushes, again painting on the layer mask as needed - IN FACT the only cuts where I didn't rely on convenient backgrounds were just me tediously tracing the entire character with a hard round brush, no shortcuts involved.

AND FINALLY