By the way, the first heat kernel proof was given - if I am not mistaken - by Peter Gilkey!
The following MSRI workshop also had some introductory remarks about index theory
I personally would recommend Gerd Grubbs first lecture; what you are looking for is on slide 5 and onwards (up to slide 7) http://jessica2.msri.org/attachments/12917/pages/129170005.htm
Paul Loya's lectures are also interesting, by the way!
Gilkey writes in
http://mmf.ruc.dk/~Booss/recoll.pdf (an interesting article, you should read it!)
"During the course, he said that there was this wonderful invariant that Bob Seeley had constructed analytically and, ‘here is Bott’s proof of the index theorem, and somebody should actually show that this gives a heat equation proof of the index theorem.’ That struck me as a fun problem, so I went home that night and gave the heat equation proof to the Gauss-Bonnet-Theorem."