Edited this answer according to your requirements:
Use CamStudio Portable to record your screen activities to AVI.
Use VirtualDubMod to convert the AVI to a sequence of images (PNG).
Use Easy Graphic Converter to convert the PNGs to GIFs.
Use UnFREEz to create an animated GIF from the images.
Done and dusted.
ALL programs are free, ALL programs but Easy Graphic Converter are portable. Either you find an alternative image converter (there are plenty, I only chose it because it was used for the tutorial) or you make it portable yourself by running the installer through the mighty Universal Extractor (I have tested it, it works).
If you want it "All in One", then you're pretty much out of luck, at least as far as "free and portable" goes.
Here are some (rather cheap) commercial screen-recording programs with output to animated GIF:
aSkysoft Screen Recorder ($39.95)
Super Screen Capture ($49.95)
As far as 'portability' is concerned, pretty much anything can be made portable, one way or other.