Since I wrote this, I thought someone might find it useful, so I share it with you:

Notes from a talk I gave in a student's seminar in Freiburg:
Komplexe K-Theorie: Bott-Periodizitaet (complex K-Theory: Bott periodicity)

It's almost the same proof as in Aityah's nice book, but maybe a little bit more lengthy (and in german).

... right now I'm sitting in the library at the ICTP in Miramare/Trieste, Italy. It's a beautiful place, and I think I'll write something about the summer school on Hodge Theory here, soon!